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Regularity properties of monotone measure-preserving maps

Regularity properties of monotone measure-preserving maps 1IntroductionGiven a pair of atom-free Borel probability measures μ\mu and ν\nu on R{\mathbb{R}}, the monotone rearrangement theorem asserts that the function y=y(x)y=y\left(x)defined implicitly by ∫−∞xdμ=∫−∞y(x)dν\underset{-\infty }{\overset{x}{\int }}{\rm{d}}\mu =\underset{-\infty }{\overset{y\left(x)}{\int }}{\rm{d}}\nu is measure preserving, i.e., μ(y−1(E))=ν(E)for any Borel setE⊂R.\mu ({y}^{-1}\left(\pmb{\mathscr{E}}))=\nu \left(\pmb{\mathscr{E}})\hspace{1.0em}\hspace{0.1em}\text{for any Borel set}\hspace{0.1em}\hspace{0.33em}\pmb{\mathscr{E}}\subset {\mathbb{R}}.In particular, yyis unique μ\mu almost everywhere and can be made to be monotone, or, equivalently, the derivative of a convex function.When R{\mathbb{R}}is replaced by Rn{{\mathbb{R}}}^{n}, however, before Brenier’s discovery [1], a proper generalization of the monotone rearrangement theorem was myth, and only after the work of McCann, in [12], was the myth made real. Precisely, he proved that if μ\mu vanishes on every Lipschitz (n−1)\left(n-1)-dimensional surface,McCann actually assumes that μ\mu vanishes on all (Borel) sets of Hausdorff dimension n−1n-1. This guarantees that the set of nondifferentiability points of a convex function are μ\mu -negligible. However, this assumption can be weakened. Since convex functions are differentiable outside of a countable union of Lipschitz hypersurfaces [14], McCann’s theorem holds assuming that μ\mu vanishes on every Lipschitz (n−1)\left(n-1)-dimensional surface.then a convex potential u:Rn→R∪{+∞}u:{{\mathbb{R}}}^{n}\to {\mathbb{R}}\cup \left\{+\infty \right\}exists whose gradient map ∇u=∇u(x)\nabla u=\nabla u\left(x)is unique μ\mu almost everywhere and pushes μ\mu forward to ν\nu , i.e., μ((∇u)−1(E))=ν(E)for any Borel setE⊂Rn.\mu \left({\left(\nabla u)}^{-1}\left(\pmb{\mathscr{E}}))=\nu \left(\pmb{\mathscr{E}})\hspace{1.0em}\hspace{0.1em}\text{for any Borel set}\hspace{0.1em}\hspace{0.33em}\pmb{\mathscr{E}}\subset {{\mathbb{R}}}^{n}.(Brenier’s theorem guaranteed the same conclusion as McCann’s theorem, but under some restrictive technical conditions on μ\mu and ν\nu .)The first general regularity result on these Brenier-McCann maps was proved by Caffarelli in [4]: provided that μ\mu and ν\nu are absolutely continuous with respect to nn-dimensional Lebesgue measure Ln{{\mathscr{L}}}^{n}, their respective densities ffand ggvanish outside of and are bounded away from zero and infinity on open bounded sets X\pmb{\mathscr{X}}and Y\pmb{\mathscr{Y}}, respectively, and Y\pmb{\mathscr{Y}}is convex, he showed that uuis strictly convex in X\pmb{\mathscr{X}}(see, e.g., the proof of [7, Theorem 4.6.2]). This result opened the door to the development of a regularity theory for mappings with convex potentials based on the regularity theory for strictly convex solutions to the Monge-Ampère equation [2]; indeed, (∇u)#f=gis formally, at least, equivalent todetD2u=fg(∇u).{\left(\nabla u)}_{\#}f=g\hspace{0.33em}\hspace{0.1em}\text{is formally, at least, equivalent to}\hspace{0.1em}\hspace{0.33em}\det {\pmb{\mathscr{D}}}^{2}u=\frac{f}{g\left(\nabla u)}.Unfortunately, Caffarelli’s boundedness assumptions on the domains X\pmb{\mathscr{X}}and Y\pmb{\mathscr{Y}}are restrictive, since many probability densities, especially those found in applications, are supported on all of Rn{{\mathbb{R}}}^{n}: Gaussian densities, for example. Motivated by this, in [5], Cordero-Erausquin and Figalli showed that, in several situations of interest, one can ensure the regularity of monotone measure-preserving maps even if the measures under consideration have unbounded supports. However, missing from their collection is the situation where Y\pmb{\mathscr{Y}}is an arbitrary convex domain. Lifting this restriction is a main goal of this article.1.1ResultsOur main theorem is an extension of Caffarelli’s theorem, on the strict convexity of uu, in two ways. First, we allow X\pmb{\mathscr{X}}and Y\pmb{\mathscr{Y}}to be unbounded. Second, we permit X\pmb{\mathscr{X}}and Y\pmb{\mathscr{Y}}to carrying certain invariant measures that we call locally doubling measures (qualitatively, our notion replaces balls in the classical notion of a doubling measure with ellipsoids, in order to account for the affine invariance of our setting).Definition 1.1A nonnegative measure λ\lambda is locally doubling (on ellipsoids) if the following holds: for every ball B\pmb{\mathscr{B}}, there is a constant C≥1\pmb{\mathscr{C}}\ge 1such that λ(ℰ)≤Cλ12ℰ\lambda \left({\mathcal{ {\mathcal E} }})\le \pmb{\mathscr{C}}\lambda \left(\frac{1}{2}{\mathcal{ {\mathcal E} }}\right)for all ellipsoids ℰ⊂B{\mathcal{ {\mathcal E} }}\subset \pmb{\mathscr{B}}with center (of mass) in spt(λ){\rm{spt}}\left(\lambda ). Here, 12ℰ\frac{1}{2}{\mathcal{ {\mathcal E} }}is the dilation of ℰ{\mathcal{ {\mathcal E} }}with respect to its center by 1/2.This notion of doubling was introduced by Jhaveri and Savin in [11].This family of measures is strictly larger than the family of measures locally comparable to Lebesgue measure on their supports. (See [11] for examples of locally doubling measures not comparable to Lebesgue measure on their supports.)That said, the first consideration of measures with a “doubling-like” property in the world of solutions to Monge-Ampère equations can be traced back to the work of Jerison [9] and then Caffarelli [3]. In particular, in [3], Caffarelli showed that Alexandrov solutions to detD2v=ρ,\det {\pmb{\mathscr{D}}}^{2}v=\rho ,where the measure ρ\rho is doubling on a specific collection of convex sets called sectionsThese are sets of the form {v≤ℓ}\left\{v\le \ell \right\}for any affine function ℓ\ell ., share the same geometric properties as Alexandrov solutions to Monge-Ampère equations with right-hand sides comparable to Lebesgue measure [2].We now state our main theorem.Theorem 1.2Let μ\mu and ν\nu be two locally doubling probability measures on Rn{{\mathbb{R}}}^{n}that vanish on Lipschitz (n−1)\left(n-1)-dimensional surfaces and are concentrated on two open sets X\pmb{\mathscr{X}}and Y\pmb{\mathscr{Y}}, respectively, and suppose that Y\pmb{\mathscr{Y}}is convex. Then any convex potential u associated to the Brenier-McCann map pushing μ\mu forward to ν\nu is strictly convex in X\pmb{\mathscr{X}}.Remark 1.3It is well known that Y\pmb{\mathscr{Y}}needs to be convex. When Y\pmb{\mathscr{Y}}is not convex, uucan fail to be strictly convex and ∇u\nabla ucan behave rather poorly. If we consider Pogorelov’s counterexample to the strict convexity of solutions to the Monge-Ampère equation in three dimensions, we see that Y\pmb{\mathscr{Y}}needs to be convex in order to guarantee the strict convexity of potentials of Brenier-McCann maps. In particular, let u(x′,x3)=∣x′∣4/3(1+x32)u\left(x^{\prime} ,{x}_{3})=| x^{\prime} {| }^{4\text{/}3}\left(1+{x}_{3}^{2}), let Qr={∣xi∣<r/2fori=1,2,3}{\pmb{\mathscr{Q}}}_{r}=\left\{| {x}_{i}| \lt r\hspace{0.1em}\text{/}\hspace{0.1em}2\hspace{0.33em}\hspace{0.1em}\text{for}\hspace{0.1em}\hspace{0.33em}i=1,2,3\right\}be the cube with side length r>0r\gt 0and centered at the origin in R3{{\mathbb{R}}}^{3}, and let Y=∇u(Qr)\pmb{\mathscr{Y}}=\nabla u\left({\pmb{\mathscr{Q}}}_{r}). If r≪1r\ll 1, then f=detD2uf=\det {\pmb{\mathscr{D}}}^{2}uis analytic and positive in Qr{\pmb{\mathscr{Q}}}_{r}, and ∇u\nabla uis the Brenier-McCann map pushing forward μ=(f/‖f‖L1(Qr))L3∣__Qr\mu =(f\hspace{0.1em}\text{/}\hspace{0.1em}\Vert f{\Vert }_{{\pmb{\mathscr{L}}}^{1}\left({\pmb{\mathscr{Q}}}_{r})}){{\mathscr{L}}}^{3}| \hspace{-0.3em}\hspace{0.1em}\text{\_\_}\hspace{0.1em}{\pmb{\mathscr{Q}}}_{r}to ν=(1/L3(Y))L3∣__Y\nu =\left(1\hspace{0.1em}\text{/}\hspace{0.1em}{{\mathscr{L}}}^{3}\left(\pmb{\mathscr{Y}})){{\mathscr{L}}}^{3}| \hspace{-0.3em}\hspace{0.1em}\text{\_\_}\hspace{0.1em}\pmb{\mathscr{Y}}. The set Y\pmb{\mathscr{Y}}is open but not convex, and u=0u=0along {x′=0}\left\{x^{\prime} =0\right\}. Moreover, as demonstrated in, for instance, [4,10], ∇u\nabla ueasily fails to be continuous when Y\pmb{\mathscr{Y}}is not convex.Remark 1.4While the target measure ν\nu need not vanishes on all Lipschitz (n−1)\left(n-1)-dimensional surfaces in order to invoke McCann’s theorem (which asks this only of the source measure μ\mu ), it must in order to ensure that our main theorem holds. If μ=L2∣__Q1\mu ={{\mathscr{L}}}^{2}| \hspace{-0.3em}\hspace{0.1em}\text{\_\_}\hspace{0.1em}{\pmb{\mathscr{Q}}}_{1}is the uniform measure on Q1{\pmb{\mathscr{Q}}}_{1}the unit cube centered at the origin in R2{{\mathbb{R}}}^{2}and ν=H1∣__Q1∩{x2=0}\nu ={{\mathscr{H}}}^{1}| \hspace{-0.3em}\hspace{0.1em}\text{\_\_}\hspace{0.1em}{\pmb{\mathscr{Q}}}_{1}\cap \left\{{x}_{2}=0\right\}is the 1-dimensional Hausdorff measure restricted to the central horizontal axis of Q1{\pmb{\mathscr{Q}}}_{1}, then the Brenier-McCann map pushing μ\mu forward to ν\nu is the projection map (x1,x2)↦x1\left({x}_{1},{x}_{2})\mapsto {x}_{1}. Up to a constant, this map’s convex potential is 12x12\frac{1}{2}{x}_{1}^{2}, which is not strictly convex. With respect our proof of Theorem 1.2, asking this of both μ\mu and ν\nu guarantees the validity of the mass balance formula in Lemma 2.1 (see also Remark 2.2), our main tool.Remark 1.5A simple case to which Theorem 1.2 applies, but the corresponding results in [4, 5] do not, is when μ=gL3∣__{∣x1∣<1}×R2\mu =g{{\mathscr{L}}}^{3}| \hspace{-0.3em}\hspace{0.1em}\text{\_\_}\hspace{0.1em}\left\{| {x}_{1}| \lt 1\right\}\times {{\mathbb{R}}}^{2}and ν=gL3∣__R2×{∣x3∣<1}\nu =g{{\mathscr{L}}}^{3}| \hspace{-0.3em}\hspace{0.1em}\text{\_\_}\hspace{0.1em}{{\mathbb{R}}}^{2}\times \left\{| {x}_{3}| \lt 1\right\}, and ggis the standard Gaussian density on R3{{\mathbb{R}}}^{3}appropriately normalized to make μ\mu and ν\nu probability measures.With our main theorem in hand, our second and third theorems further extend the known regularity theory for monotone measure-preserving maps, completing the story started by Cordero-Erausquin and Figalli in [5] on monotone transports between unbounded domains.Theorem 1.6Let μ\mu and ν\nu be two locally doubling probability measures on Rn{{\mathbb{R}}}^{n}that vanish on Lipschitz (n−1)\left(n-1)-dimensional surfaces and are concentrated on two open sets X\pmb{\mathscr{X}}and Y\pmb{\mathscr{Y}}, respectively, and suppose that Y\pmb{\mathscr{Y}}is convex. Then the Brenier-McCann map ∇u\nabla upushing μ\mu forward to ν\nu is a homeomorphism from X\pmb{\mathscr{X}}onto a full measure subset of Y\pmb{\mathscr{Y}}. Moreover, for every A⋐X\pmb{\mathscr{A}}\hspace{0.33em}\Subset \hspace{0.33em}\pmb{\mathscr{X}}, a constant α>0\alpha \gt 0exists such that ∇u∈C0,α(A)\nabla u\in {\pmb{\mathscr{C}}}^{0,\alpha }\left(\pmb{\mathscr{A}}). Furthermore, ∇u(X)=Y\nabla u\left(\pmb{\mathscr{X}})=\pmb{\mathscr{Y}}whenever X\pmb{\mathscr{X}}is convex.Theorem 1.7Let f and g be two functions on Rn{{\mathbb{R}}}^{n}that define locally doubling probability measures concentrated on two open sets X\pmb{\mathscr{X}}and Y\pmb{\mathscr{Y}}, respectively, and suppose that Y\pmb{\mathscr{Y}}is convex. Assume that f and g are bounded away from zero and infinity on compact subsets of X\pmb{\mathscr{X}}and Y\pmb{\mathscr{Y}}, respectively. Then for every E⋐X\pmb{\mathscr{E}}\hspace{0.33em}\Subset \hspace{0.33em}\pmb{\mathscr{X}}, a constant ε>0\varepsilon \gt 0exists such that any convex potential u associated to the Brenier-McCann map pushing f forward to g is W2,1+ε(E){\pmb{\mathscr{W}}}^{2,1+\varepsilon }\left(\pmb{\mathscr{E}}). Also, ∇u\nabla uis locally a Ck+1,β{\pmb{\mathscr{C}}}^{k+1,\beta }-diffeomorphism from X\pmb{\mathscr{X}}onto its image provided f and g are locally Ck,β{\pmb{\mathscr{C}}}^{k,\beta }in X\pmb{\mathscr{X}}and Y\pmb{\mathscr{Y}}, respectively.Remark 1.8We note that the proof of the Theorem 1.7, given the strict convexity of uu(provided by Theorem 1.2), is classical. Indeed, it suffices to localize classical regularity results for the Monge-Ampère equation. We refer the reader to [7, Section 4.6.1] for more details.1.2StructureThis remainder of this article is structured as follows.In Section 2, we prove Theorem 1.2. Our proof is self-contained apart from some facts in convex analysis; we provide explicit references to these used but unproved facts. We remark that our proof is inspired by the proof of the Alexandrov maximum principle in [11] (and, of course, Caffarelli’s original proof of the strict convexity of potential functions of optimal transports/solutions to Monge-Ampère equations). If the reader is familiar with [5] or [4], then they might consider directing their attention to Case 2. Case 2b is completely novel. Case 2a illustrates our argument in the setting of [4], which builds on the work of [2] and is the foundation for Case 2b.Section 3 is dedicated to the proof of Theorem 1.6. Our proof here is similarly self-contained (and an adaptation of Caffarelli’s argument of the same result in [2], but, of course, using the line of reasoning developed to prove Theorem 1.2). The Hölder regularity of ∇u\nabla uis a consequence of appropriately localizing the arguments of [11].2Proof of Theorem 1.2Before we begin, it will be convenient to replace the potential uuby the following lower-semicontinuous extension of uuoutside of X\pmb{\mathscr{X}}:Here, ∂u(z)\partial u\left(z)is called the subdifferential of uuat zzand is defined as follows: ∂u(z)≔{p∈Rn:u(x)≥u(z)+p⋅(x−z)for allx∈X}.\partial u\left(z):= \left\{p\in {{\mathbb{R}}}^{n}:u\left(x)\ge u\left(z)+p\cdot \left(x-z)\hspace{0.4em}\text{for all}\hspace{0.4em}\hspace{0.1em}x\in \pmb{\mathscr{X}}\text{}\right\}.Moreover, for a set E⊂Rn\pmb{\mathscr{E}}\subset {{\mathbb{R}}}^{n}, we define ∂u(E)≔∪z∈E∂u(z)\partial u\left(\pmb{\mathscr{E}}):= {\cup }_{z\in \pmb{\mathscr{E}}}\partial u\left(z).u̲(x)≔supz∈Xp∈∂u(z){u(z)+p⋅(x−z)}.\underline{u}\left(x):= \mathop{\sup }\limits_{\begin{array}{c}z\in \pmb{\mathscr{X}}\\ p\in \partial u\left(z)\end{array}}\left\{u\left(z)+p\cdot \left(x-z)\right\}.Observe that u̲∣X=u∣X\underline{u}{| }_{\pmb{\mathscr{X}}}=u{| }_{\pmb{\mathscr{X}}}. For notational simplicity, we shall not distinguish u̲\underline{u}from uu; so when we write uuin what follows, we mean u̲\underline{u}.We shall denote the domain of uuby dom(u){\rm{dom}}\left(u), namely, dom(u)≔{u<+∞}{\rm{dom}}\left(u):= \left\{u\lt +\infty \right\}. Note that dom(u){\rm{dom}}\left(u)is convex. We recall that convex functions are locally Lipschitz inside their domain [7, Appendix A.4]. Furthermore, we shall denote the convex hull of a set A\pmb{\mathscr{A}}by conv(A){\rm{conv}}\left(\pmb{\mathscr{A}}).Let ℓ\ell define a supporting plane to the graph of uuat a point in X\pmb{\mathscr{X}}. Precisely, ℓ(x)=u(z)+p⋅(x−z)for some(z,p)∈X×Rn\ell \left(x)=u\left(z)+p\cdot \left(x-z)\hspace{1.0em}\hspace{0.1em}\text{for some}\hspace{0.1em}\hspace{0.33em}\left(z,p)\in \pmb{\mathscr{X}}\times {{\mathbb{R}}}^{n}and ℓ≤u\ell \le u. Note that Σ≔{u=ℓ}={u≤ℓ}\Sigma := \left\{u=\ell \right\}=\left\{u\le \ell \right\}is closed, as uuis lower-semicontinuous, Σ,X⊂dom(u),\Sigma ,\pmb{\mathscr{X}}\subset {\rm{dom}}\left(u),and, because Y\pmb{\mathscr{Y}}is convex, (2.1)∂u(Rn)⊂∂u(X)¯⊂Y¯andLn(Y⧹∂u(X))=0.\partial u\left({{\mathbb{R}}}^{n})\subset \overline{\partial u\left(\pmb{\mathscr{X}})}\subset \overline{\pmb{\mathscr{Y}}}\hspace{1.0em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1.0em}{{\mathscr{L}}}^{n}\left(\pmb{\mathscr{Y}}\setminus \partial u\left(\pmb{\mathscr{X}}))=0.(A proof of (2.1) can be found in [5].)Recall that an exposed point xˆ\hat{x}of Σ⊂Rn\Sigma \subset {{\mathbb{R}}}^{n}is one for which there exists a hyperplane Π⊂Rn\Pi \subset {{\mathbb{R}}}^{n}tangent to Σ\Sigma at xˆ\hat{x}such that Π∩Σ={xˆ}\Pi \cap \Sigma =\left\{\hat{x}\right\}. Also, remember that optimal/monotone transports balance mass, in the following way.Lemma 2.1(Mass balance formula) Let u:Rn→R∪{+∞}u:{{\mathbb{R}}}^{n}\to {\mathbb{R}}\cup \left\{+\infty \right\}be convex and such that (∇u)#μ=ν{\left(\nabla u)}_{\#}\mu =\nu where μ\mu and ν\nu are two Borel measures that vanish on all Lipschitz (n−1)\left(n-1)-dimensional surfaces. Then for all Borel sets E⊂Rn\pmb{\mathscr{E}}\subset {{\mathbb{R}}}^{n}, μ(E)=ν(∂u(E)).\mu \left(\pmb{\mathscr{E}})=\nu \left(\partial u\left(\pmb{\mathscr{E}})).Remark 2.2The mass balance formula was originally proved for measures that are absolutely continuous with respect to Lebesgue measure [13, Lemma 4.6]. However, with respect to absolute continuity, the proof only relies on the measures in question not giving mass to the set of nondifferentiable points of a convex function. As observed in Section 1, such points are contained in a countable union of Lipschitz (n−1)\left(n-1)-dimensional surfaces. So the set of nondifferentiable points of a convex function is negligible both for μ\mu and ν\nu under our assumption.Finally, recall that if a nonnegative measure is locally doubling (on ellipsoids), then it is locally doubling on all bounded convex domains [11, Corollary 2.5]. After this preliminary discussion, we can now prove our main theorem.Proof of Theorem 1.2Proving that uuis strictly convex in X\pmb{\mathscr{X}}corresponds to proving that for any supporting plane ℓ\ell to (the graph of) uuat a point in X\pmb{\mathscr{X}}, the set Σ={u=ℓ}\Sigma =\left\{u=\ell \right\}is a singleton. Assuming that Σ\Sigma is not a singleton, we will show that Σ\Sigma both has and does not have exposed points, which cannot be; thus, Σ\Sigma is a singleton, as desired.Case 1. Σ\Sigma has no exposed points. If Σ\Sigma has no exposed points, then Σ⊃Re\Sigma \supset {\mathbb{R}}{\bf{e}}for some unit vector e{\bf{e}}. In turn, ∂u(Rn)⊂e⊥\partial u\left({{\mathbb{R}}}^{n})\subset {{\bf{e}}}^{\perp }[7, Lemma A.25]. But this is impossible given (2.1): 0<Ln(∂u(X)∩Y)≤Ln(∂u(Rn)∩Y)≤Ln(e⊥∩Y)=0.0\lt {{\mathscr{L}}}^{n}\left(\partial u\left(\pmb{\mathscr{X}})\cap \pmb{\mathscr{Y}})\le {{\mathscr{L}}}^{n}\left(\partial u\left({{\mathbb{R}}}^{n})\cap \pmb{\mathscr{Y}})\le {{\mathscr{L}}}^{n}\left({{\bf{e}}}^{\perp }\cap \pmb{\mathscr{Y}})=0.Case 2. Σ\Sigma has an exposed point xˆ\hat{x}in X¯\overline{\pmb{\mathscr{X}}}. Up to a translation and a rotation, we can assume that xˆ=0∈X¯,Σ⊂{x1≤0},andΣ∩{x1=0}={0}.\hat{x}=0\in \overline{\pmb{\mathscr{X}}},\hspace{1.0em}\Sigma \subset \left\{{x}_{1}\le 0\right\},\hspace{1.0em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1.0em}\Sigma \cap \left\{{x}_{1}=0\right\}=\left\{0\right\}.Since X\pmb{\mathscr{X}}is open and Σ∩X\Sigma \cap \pmb{\mathscr{X}}is nonempty by construction, there is a point xint∈Σ∩X{x}_{int}\in \Sigma \cap \pmb{\mathscr{X}}and a ball centered at this point completely contained in X\pmb{\mathscr{X}}. Thus, up to a shearing transformation x↦x−ηx1x\mapsto x-\eta {x}_{1}with η⋅e1=0\eta \cdot {{\bf{e}}}_{1}=0, and a dilation, we may assume that xint=−e1andBd(−e1)⋐X{x}_{int}=-{{\bf{e}}}_{1}\hspace{1.0em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1.0em}{\pmb{\mathscr{B}}}_{d}\left(-{{\bf{e}}}_{1})\hspace{0.33em}\Subset \hspace{0.33em}\pmb{\mathscr{X}}for some d>0d\gt 0. Finally, up to subtracting ℓ\ell from uu, we can assume that ℓ≡0.\ell \equiv 0.Case 2a: 0∈int(dom(u))∩X¯0\in {\rm{int}}\left({\rm{dom}}\left(u))\cap \overline{\pmb{\mathscr{X}}}. As our exposed point 0 and all of the points in Bd(−e1)¯\overline{{\pmb{\mathscr{B}}}_{d}\left(-{{\bf{e}}}_{1})}belong to int(dom(u)){\rm{int}}\left({\rm{dom}}\left(u)), which is convex (and, by definition, open), the convex hull of the union of Bd(−e1)¯\overline{{\pmb{\mathscr{B}}}_{d}\left(-{{\bf{e}}}_{1})}and {0}\left\{0\right\}is contained in int(dom(u)){\rm{int}}\left({\rm{dom}}\left(u)). So there exists an open, bounded set U⋐int(dom(u))\pmb{\mathscr{U}}\hspace{0.33em}\Subset \hspace{0.33em}{\rm{int}}\left({\rm{dom}}\left(u))containing conv(Bd(−e1)¯∪{0}){\rm{conv}}\left(\overline{{\pmb{\mathscr{B}}}_{d}\left(-{{\bf{e}}}_{1})}\cup \left\{0\right\}). Moreover, we know that ∂u(U)⊂conv(∇u(U)¯)≕ϒ⊂BR∩Y¯\partial u\left(\pmb{\mathscr{U}})\subset {\rm{conv}}\left(\overline{\nabla u\left(\pmb{\mathscr{U}})})\hspace{0.33em}=: \hspace{0.33em}\Upsilon \subset {\pmb{\mathscr{B}}}_{\pmb{\mathscr{R}}}\cap \overline{\pmb{\mathscr{Y}}}for some R>0\pmb{\mathscr{R}}\gt 0[7, Lemma A.22]. Let u∗{u}^{\ast }be the Legendre transform of uu, namely, (2.2)u∗(q)≔supx∈Rn{q⋅x−u(x)}{u}^{\ast }\left(q):= \mathop{\sup }\limits_{x\in {{\mathbb{R}}}^{n}}\left\{q\cdot x-u\left(x)\right\}and define Ω≔∂u∗(ϒ)⊃U.\Omega := \partial {u}^{\ast }\left(\Upsilon )\supset \pmb{\mathscr{U}}.Recalling that ∂u\partial uand ∂u∗\partial {u}^{\ast }are inverses of each other [7, Section A.4.2], we deduce that (∇u)#ρ=γ{\left(\nabla u)}_{\#}\rho =\gamma , where ρ≔μ∣__Ωandγ≔ν∣__ϒ.\rho := \mu | \hspace{-0.3em}\hspace{0.1em}\text{\_\_}\hspace{0.1em}\Omega \hspace{1.0em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1.0em}\gamma := \nu | \hspace{-0.3em}\hspace{0.1em}\text{\_\_}\hspace{0.1em}\Upsilon .In particular, if we let ϕ:Rn→R∪{+∞}\phi :{{\mathbb{R}}}^{n}\to {\mathbb{R}}\cup \left\{+\infty \right\}be defined by ϕ(x)≔supz∈Ωp∈∂u(z){u(z)+p⋅(x−z)},\phi \left(x):= \mathop{\sup }\limits_{\begin{array}{c}z\in \Omega \\ p\in \partial u\left(z)\end{array}}\left\{u\left(z)+p\cdot \left(x-z)\right\},then, by construction, ϕ\phi and uuagree on Ω\Omega , ϕ\phi is (globally) Lipschitz, ∂ϕ(Rn)=∂ϕ(Ω)=ϒ,\partial \phi \left({{\mathbb{R}}}^{n})=\partial \phi \left(\Omega )=\Upsilon ,and 0∈Ω0\in \Omega is an exposed point for {ϕ=0}={ϕ≤0}\left\{\phi =0\right\}=\left\{\phi \le 0\right\}.Now, let ϕε(x)≔ϕ(x)−ε(x1+1){\phi }_{\varepsilon }\left(x):= \phi \left(x)-\varepsilon \left({x}_{1}+1)and define S0≔{ϕ=0}∩{x1≥−1}andSε≔{ϕε≤0}.{\pmb{\mathscr{S}}}_{0}:= \left\{\phi =0\right\}\cap \left\{{x}_{1}\ge -1\right\}\hspace{1.0em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1.0em}{\pmb{\mathscr{S}}}_{\varepsilon }:= \left\{{\phi }_{\varepsilon }\le 0\right\}.Also, let γε{\gamma }_{\varepsilon }be defined by γε≔(Id−εe1)#γ.{\gamma }_{\varepsilon }:= {\left({\rm{Id}}-\varepsilon {{\bf{e}}}_{1})}_{\#}\gamma .Notice that, by construction, S0{\pmb{\mathscr{S}}}_{0}is compact, S0⊂{x1≤0}{\pmb{\mathscr{S}}}_{0}\subset \left\{{x}_{1}\le 0\right\}, 0,−e1∈S00,-{{\bf{e}}}_{1}\in {\pmb{\mathscr{S}}}_{0}, and Sε→S0{\pmb{\mathscr{S}}}_{\varepsilon }\to {\pmb{\mathscr{S}}}_{0}in the Hausdorff sense as ε→0\varepsilon \to 0; in particular, there exists D>0\pmb{\mathscr{D}}\gt 0such that Sε⊂BDfor allε≪1.{\pmb{\mathscr{S}}}_{\varepsilon }\subset {\pmb{\mathscr{B}}}_{\pmb{\mathscr{D}}}\hspace{1.0em}\hspace{0.1em}\text{for all}\hspace{0.1em}\hspace{0.33em}\varepsilon \ll 1.Also, if aε>0{a}_{\varepsilon }\gt 0is such that Πε≔{x1=aε}{\Pi }_{\varepsilon }:= \left\{{x}_{1}={a}_{\varepsilon }\right\}is a supporting plane to Sε{\pmb{\mathscr{S}}}_{\varepsilon }, we see that aε→0asε→0{a}_{\varepsilon }\to 0\hspace{1.0em}\hspace{0.1em}\text{as}\hspace{0.1em}\hspace{0.33em}\varepsilon \to 0and ε=∣ϕε(0)∣≤maxSε∣ϕε∣≤(1+aε)ε.\varepsilon =| {\phi }_{\varepsilon }\left(0)| \le \mathop{\max }\limits_{{\pmb{\mathscr{S}}}_{\varepsilon }}| {\phi }_{\varepsilon }| \le \left(1+{a}_{\varepsilon })\varepsilon .Let Aε{\pmb{\mathscr{A}}}_{\varepsilon }be the John transformation (affine map) that normalizes Sε{\pmb{\mathscr{S}}}_{\varepsilon }[6]: Aεx≔Lε(x−xε),{\pmb{\mathscr{A}}}_{\varepsilon }x:= {\pmb{\mathscr{L}}}_{\varepsilon }\left(x-{x}_{\varepsilon }),where xε{x}_{\varepsilon }is the center of mass of Sε{\pmb{\mathscr{S}}}_{\varepsilon }and Lε:Rn→Rn{\pmb{\mathscr{L}}}_{\varepsilon }:{{\mathbb{R}}}^{n}\to {{\mathbb{R}}}^{n}is a symmetric and positive definite linear transformation. Set ϕ˜ε(x)≔ϕε(Aε−1x)εandS˜ε≔Aε(Sε).{\tilde{\phi }}_{\varepsilon }\left(x):= \frac{{\phi }_{\varepsilon }\left({\pmb{\mathscr{A}}}_{\varepsilon }^{-1}x)}{\varepsilon }\hspace{1.0em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1.0em}{\tilde{\pmb{\mathscr{S}}}}_{\varepsilon }:= {\pmb{\mathscr{A}}}_{\varepsilon }\left({\pmb{\mathscr{S}}}_{\varepsilon }).Then B1⊂S˜ε⊂Bn3/2{\pmb{\mathscr{B}}}_{1}\subset {\tilde{\pmb{\mathscr{S}}}}_{\varepsilon }\subset {\pmb{\mathscr{B}}}_{{n}^{3\text{/}2}}and 1=∣ϕ˜ε(0˜ε)∣≤maxS˜ε∣ϕ˜ε∣≤1+aεwith0˜ε≔Aε(0).1=| {\tilde{\phi }}_{\varepsilon }\left({\tilde{0}}_{\varepsilon })| \le \mathop{\max }\limits_{{\tilde{\pmb{\mathscr{S}}}}_{\varepsilon }}| {\tilde{\phi }}_{\varepsilon }| \le 1+{a}_{\varepsilon }\hspace{1.0em}\hspace{0.1em}\text{with}\hspace{0.1em}\hspace{0.33em}{\tilde{0}}_{\varepsilon }:= {\pmb{\mathscr{A}}}_{\varepsilon }\left(0).Recall that affine transformations preserve the ratio of the distances between parallel planes; therefore, letting Π−1≔{x1=−1}{\Pi }_{-1}:= \left\{{x}_{1}=-1\right\}, Π0≔{x1=0}{\Pi }_{0}:= \left\{{x}_{1}=0\right\}, and Π˜i≔Aε(Πi){\tilde{\Pi }}_{i}:= {\pmb{\mathscr{A}}}_{\varepsilon }\left({\Pi }_{i})for i=−1,0,εi=-1,0,\varepsilon , we have that dist(Π˜0,Π˜ε)dist(Π˜−1,Π˜ε)=dist(Π0,Πε)dist(Π−1,Πε)=aε1+aε.\frac{{\rm{dist}}\left({\tilde{\Pi }}_{0},{\tilde{\Pi }}_{\varepsilon })}{{\rm{dist}}\left({\tilde{\Pi }}_{-1},{\tilde{\Pi }}_{\varepsilon })}=\frac{{\rm{dist}}\left({\Pi }_{0},{\Pi }_{\varepsilon })}{{\rm{dist}}\left({\Pi }_{-1},{\Pi }_{\varepsilon })}=\frac{{a}_{\varepsilon }}{1+{a}_{\varepsilon }}.In turn, dist(0˜ε,∂S˜ε)≤dist(Π˜0,Π˜ε)≤dist(Π˜−1,Π˜ε)aε1+aε≤diam(S˜ε)aε≤2n3/2aε,{\rm{dist}}\left({\tilde{0}}_{\varepsilon },\partial {\tilde{\pmb{\mathscr{S}}}}_{\varepsilon })\le {\rm{dist}}\left({\tilde{\Pi }}_{0},{\tilde{\Pi }}_{\varepsilon })\le {\rm{dist}}\left({\tilde{\Pi }}_{-1},{\tilde{\Pi }}_{\varepsilon })\frac{{a}_{\varepsilon }}{1+{a}_{\varepsilon }}\le {\rm{diam}}\left({\tilde{\pmb{\mathscr{S}}}}_{\varepsilon }){a}_{\varepsilon }\le 2{n}^{3\text{/}2}{a}_{\varepsilon },and considering the cone generated by ∂S˜ε\partial {\tilde{\pmb{\mathscr{S}}}}_{\varepsilon }over (0˜ε,ϕ˜ε(0˜ε))\left({\tilde{0}}_{\varepsilon },{\tilde{\phi }}_{\varepsilon }\left({\tilde{0}}_{\varepsilon })), we find that Kε≔convBrn∪rnaεe1⊂∂ϕ˜ε(S˜ε)withrn≔12n3/2.{\pmb{\mathscr{K}}}_{\varepsilon }:= {\rm{conv}}\left({\pmb{\mathscr{B}}}_{{r}_{n}}\cup \left\{\phantom{\rule[-1.25em]{}{0ex}},\frac{{r}_{n}}{{a}_{\varepsilon }}{{\bf{e}}}_{1}\right\}\right)\subset \partial {\tilde{\phi }}_{\varepsilon }\left({\tilde{\pmb{\mathscr{S}}}}_{\varepsilon })\hspace{1.0em}\hspace{0.1em}\text{with}\hspace{0.1em}\hspace{0.33em}{r}_{n}:= \frac{1}{2{n}^{3\text{/}2}}.(For more details on this inclusion, see, e.g., [7, Theorem 2.8].) So if we let ρ˜ε≔(Aε)#ρandγ˜ε≔(ε−1Lε−1)#γε,{\tilde{\rho }}_{\varepsilon }:= {\left({\pmb{\mathscr{A}}}_{\varepsilon })}_{\#}\rho \hspace{1.0em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1.0em}{\tilde{\gamma }}_{\varepsilon }:= {\left({\varepsilon }^{-1}{\pmb{\mathscr{L}}}_{\varepsilon }^{-1})}_{\#}{\gamma }_{\varepsilon },then (∇ϕ˜ε)#ρ˜ε=γ˜ε{\left(\nabla {\tilde{\phi }}_{\varepsilon })}_{\#}{\tilde{\rho }}_{\varepsilon }={\tilde{\gamma }}_{\varepsilon }, and, by the mass balance formula, (2.3)γ˜ε(Kε)≤γ˜ε(∂ϕ˜ε(S˜ε))=ρ˜ε(S˜ε)≤ρ˜ε(Bn3/2).{\tilde{\gamma }}_{\varepsilon }\left({\pmb{\mathscr{K}}}_{\varepsilon })\le {\tilde{\gamma }}_{\varepsilon }(\partial {\tilde{\phi }}_{\varepsilon }\left({\tilde{\pmb{\mathscr{S}}}}_{\varepsilon }))={\tilde{\rho }}_{\varepsilon }\left({\tilde{\pmb{\mathscr{S}}}}_{\varepsilon })\le {\tilde{\rho }}_{\varepsilon }\left({\pmb{\mathscr{B}}}_{{n}^{3\text{/}2}}).On the other hand, since Sε⊂BD{\pmb{\mathscr{S}}}_{\varepsilon }\subset {\pmb{\mathscr{B}}}_{\pmb{\mathscr{D}}}and S˜ε⊃B1{\tilde{\pmb{\mathscr{S}}}}_{\varepsilon }\supset {\pmb{\mathscr{B}}}_{1}for ε≪1\varepsilon \ll 1, we see that ∣Aε(w)−Aε(z)∣≥1D∣w−z∣for allw,z∈Rn.| {\pmb{\mathscr{A}}}_{\varepsilon }\left(w)-{\pmb{\mathscr{A}}}_{\varepsilon }\left(z)| \ge \frac{1}{\pmb{\mathscr{D}}}| w-z| \hspace{1.0em}\hspace{0.1em}\text{for all}\hspace{0.1em}\hspace{0.33em}w,z\in {{\mathbb{R}}}^{n}.In turn, for ε≪1\varepsilon \ll 1, Ω˜ε≔Aε(Ω)⊃Aε(Bd(−e1))⊃BdD(Aε(−e1)).{\tilde{\Omega }}_{\varepsilon }:= {\pmb{\mathscr{A}}}_{\varepsilon }\left(\Omega )\supset {\pmb{\mathscr{A}}}_{\varepsilon }\left({\pmb{\mathscr{B}}}_{d}\left(-{{\bf{e}}}_{1}))\supset {\pmb{\mathscr{B}}}_{\tfrac{d}{\pmb{\mathscr{D}}}}\left({\pmb{\mathscr{A}}}_{\varepsilon }\left(-{{\bf{e}}}_{1})).Therefore, if we define S˜ε,d≔dist(⋅,∂S˜ε)≥d2D,{\tilde{\pmb{\mathscr{S}}}}_{\varepsilon ,d}:= \left\{\phantom{\rule[-1.25em]{}{0ex}},{\rm{dist}}\left(\cdot ,\partial {\tilde{\pmb{\mathscr{S}}}}_{\varepsilon })\ge \frac{d}{2\pmb{\mathscr{D}}}\right\},then there exists a dimensional constant Cn>0{\pmb{\mathscr{C}}}_{n}\gt 0and a point z˜d{\tilde{z}}_{d}such that Aε(Bd(−e1))∩S˜ε,d⊃BdCnD(z˜d).{\pmb{\mathscr{A}}}_{\varepsilon }\left({\pmb{\mathscr{B}}}_{d}\left(-{{\bf{e}}}_{1}))\cap {\tilde{\pmb{\mathscr{S}}}}_{\varepsilon ,d}\supset {\pmb{\mathscr{B}}}_{\tfrac{d}{{\pmb{\mathscr{C}}}_{n}\pmb{\mathscr{D}}}}\left({\tilde{z}}_{d}).Also, by, for example, [7, Corollary A.23], ∂ϕ˜ε(S˜ε,d)⊂B6Dd.\partial {\tilde{\phi }}_{\varepsilon }\left({\tilde{\pmb{\mathscr{S}}}}_{\varepsilon ,d})\subset {\pmb{\mathscr{B}}}_{\tfrac{6\pmb{\mathscr{D}}}{d}}.Thus, for all ε≪1\varepsilon \ll 1, (2.4)ρ˜ε(Bn3/2)≤μ˜ε(B2n3/2(z˜d))≤Cμkμ˜εBdCnD(z˜d)=Cμkρ˜εBdCnD(z˜d),{\tilde{\rho }}_{\varepsilon }\left({\pmb{\mathscr{B}}}_{{n}^{3\text{/}2}})\le {\tilde{\mu }}_{\varepsilon }\left({\pmb{\mathscr{B}}}_{2{n}^{3\text{/}2}}\left({\tilde{z}}_{d}))\le {\pmb{\mathscr{C}}}_{\mu }^{k}{\tilde{\mu }}_{\varepsilon }\left({\pmb{\mathscr{B}}}_{\tfrac{d}{{\pmb{\mathscr{C}}}_{n}\pmb{\mathscr{D}}}}\left({\tilde{z}}_{d})\right)={\pmb{\mathscr{C}}}_{\mu }^{k}\hspace{0.16em}{\tilde{\rho }}_{\varepsilon }\left({\pmb{\mathscr{B}}}_{\tfrac{d}{{\pmb{\mathscr{C}}}_{n}\pmb{\mathscr{D}}}}\left({\tilde{z}}_{d})\right),where μ˜ε≔(Aε)#μ{\tilde{\mu }}_{\varepsilon }:= {\left({\pmb{\mathscr{A}}}_{\varepsilon })}_{\#}\mu , the number k∈Nk\in {\mathbb{N}}is such that 2n3/2≤2kdCnD2{n}^{3\text{/}2}\le {2}^{k}\frac{d}{{\pmb{\mathscr{C}}}_{n}\pmb{\mathscr{D}}}, and Cμ{\pmb{\mathscr{C}}}_{\mu }is the doubling constant for μ\mu in B4Dn3/2{\pmb{\mathscr{B}}}_{4\pmb{\mathscr{D}}{n}^{3\text{/}2}}. (The last equality holds since μ˜ε{\tilde{\mu }}_{\varepsilon }and ρ˜ε{\tilde{\rho }}_{\varepsilon }agree on Ω˜ε{\tilde{\Omega }}_{\varepsilon }.) Moreover, by using the mass balance formula again, we deduce that (2.5)ρ˜εBdCnD(z˜d)≤ρ˜ε(Aε(Bd(−e1))∩S˜ε,d)=γ˜ε(∂ϕ˜ε(Aε(Bd(−e1))∩S˜ε,d))≤γ˜εB6Dd{\tilde{\rho }}_{\varepsilon }\left({\pmb{\mathscr{B}}}_{\tfrac{d}{{\pmb{\mathscr{C}}}_{n}\pmb{\mathscr{D}}}}\left({\tilde{z}}_{d})\right)\le {\tilde{\rho }}_{\varepsilon }({\pmb{\mathscr{A}}}_{\varepsilon }\left({\pmb{\mathscr{B}}}_{d}\left(-{{\bf{e}}}_{1}))\cap {\tilde{\pmb{\mathscr{S}}}}_{\varepsilon ,d})={\tilde{\gamma }}_{\varepsilon }(\partial {\tilde{\phi }}_{\varepsilon }\left({\pmb{\mathscr{A}}}_{\varepsilon }\left({\pmb{\mathscr{B}}}_{d}\left(-{{\bf{e}}}_{1}))\cap {\tilde{\pmb{\mathscr{S}}}}_{\varepsilon ,d}))\le {\tilde{\gamma }}_{\varepsilon }\left({\pmb{\mathscr{B}}}_{\tfrac{6\pmb{\mathscr{D}}}{d}}\right)for all ε≪1\varepsilon \ll 1. Consequently, combining the three chains of inequalities (2.3), (2.4), and (2.5), we have that (2.6)γ˜ε(Kε)≤Cμkγ˜εB6Dd.{\tilde{\gamma }}_{\varepsilon }\left({\pmb{\mathscr{K}}}_{\varepsilon })\le {\pmb{\mathscr{C}}}_{\mu }^{k}{\tilde{\gamma }}_{\varepsilon }\left({\pmb{\mathscr{B}}}_{\tfrac{6\pmb{\mathscr{D}}}{d}}\right).Now, let tme1∈Kε{t}_{m}{{\bf{e}}}_{1}\in {\pmb{\mathscr{K}}}_{\varepsilon }for m=1,…,Mm=1,\ldots ,\pmb{\mathscr{M}}be a sequence of points chosenA possible way to construct such a sequence is to choose tm=5m{t}_{m}={5}^{m}. To ensure that tme1∈Kε{t}_{m}{{\bf{e}}}_{1}\in {\pmb{\mathscr{K}}}_{\varepsilon }for any m=1,…,Mm=1,\ldots ,\pmb{\mathscr{M}}, one needs M≤logrn−logaεlog5\pmb{\mathscr{M}}\le \frac{\log {r}_{n}-\log {a}_{\varepsilon }}{\log 5}.so that 12Km⊂Km⧹Km−1withKm≔conv(Brn∪{tme1})andK0≔Brn.\frac{1}{2}{\pmb{\mathscr{K}}}_{m}\subset {\pmb{\mathscr{K}}}_{m}\setminus {\pmb{\mathscr{K}}}_{m-1}\hspace{1.0em}\hspace{0.1em}\text{with}\hspace{0.1em}\hspace{0.33em}{\pmb{\mathscr{K}}}_{m}:= {\rm{conv}}\left({\pmb{\mathscr{B}}}_{{r}_{n}}\cup \left\{{t}_{m}{{\bf{e}}}_{1}\right\})\hspace{1.0em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1.0em}{\pmb{\mathscr{K}}}_{0}:= {\pmb{\mathscr{B}}}_{{r}_{n}}.By construction, 12Kmm=1M{\left\{\frac{1}{2}{\pmb{\mathscr{K}}}_{m}\right\}}_{m=1}^{\pmb{\mathscr{M}}}is a disjoint family, and (2.7)M=M(aε)→∞asaε→0.\pmb{\mathscr{M}}=\pmb{\mathscr{M}}\left({a}_{\varepsilon })\to \infty \hspace{1.0em}\hspace{0.1em}\text{as}\hspace{0.1em}\hspace{0.33em}{a}_{\varepsilon }\to 0.Hence, since εLε(Kε)⊂ϒ\varepsilon {\pmb{\mathscr{L}}}_{\varepsilon }\left({\pmb{\mathscr{K}}}_{\varepsilon })\subset \Upsilon , we find that Mγ˜ε(Brn)≤∑m=1Mγ˜ε(Km)≤Cγ∑m=1Mγ˜ε12Km≤Cγγ˜ε(Kε),\pmb{\mathscr{M}}{\tilde{\gamma }}_{\varepsilon }\left({\pmb{\mathscr{B}}}_{{r}_{n}})\le \mathop{\sum }\limits_{m=1}^{\pmb{\mathscr{M}}}{\tilde{\gamma }}_{\varepsilon }\left({\pmb{\mathscr{K}}}_{m})\le {\pmb{\mathscr{C}}}_{\gamma }\mathop{\sum }\limits_{m=1}^{\pmb{\mathscr{M}}}{\tilde{\gamma }}_{\varepsilon }\left(\frac{1}{2}{\pmb{\mathscr{K}}}_{m}\right)\le {\pmb{\mathscr{C}}}_{\gamma }{\tilde{\gamma }}_{\varepsilon }\left({\pmb{\mathscr{K}}}_{\varepsilon }),with Cγ{\pmb{\mathscr{C}}}_{\gamma }denoting the doubling constant for γ\gamma in B2R{\pmb{\mathscr{B}}}_{2\pmb{\mathscr{R}}}, which is the same as the doubling constant for ν\nu in B2R{\pmb{\mathscr{B}}}_{2\pmb{\mathscr{R}}}; since ϒ\Upsilon is convex, γ\gamma inherits its doubling property from ν\nu . All in all, considering the aforementioned chain of inequalities and (2.6) and denoting by j∈Nj\in {\mathbb{N}}the smallest number such that 6Dd≤2jrn\frac{6\pmb{\mathscr{D}}}{d}\le {2}^{j}{r}_{n}, we see that 0<Mγ˜ε(Brn)≤CμkCγγ˜εB6Dd≤CμkCγj+1γ˜ε(Brn),0\lt \pmb{\mathscr{M}}{\tilde{\gamma }}_{\varepsilon }\left({\pmb{\mathscr{B}}}_{{r}_{n}})\le {\pmb{\mathscr{C}}}_{\mu }^{k}{\pmb{\mathscr{C}}}_{\gamma }{\tilde{\gamma }}_{\varepsilon }\left({\pmb{\mathscr{B}}}_{\tfrac{6\pmb{\mathscr{D}}}{d}}\right)\le {\pmb{\mathscr{C}}}_{\mu }^{k}{\pmb{\mathscr{C}}}_{\gamma }^{j+1}{\tilde{\gamma }}_{\varepsilon }\left({\pmb{\mathscr{B}}}_{{r}_{n}}),or, equivalently, M≤CμkCγj+1.\pmb{\mathscr{M}}\le {\pmb{\mathscr{C}}}_{\mu }^{k}{\pmb{\mathscr{C}}}_{\gamma }^{j+1}.But this is impossible for small ε\varepsilon , concluding the proof.Case 2b: 0∈∂(dom(u))∩X¯0\in \partial \left({\rm{dom}}\left(u))\cap \overline{\pmb{\mathscr{X}}}. In this subcase, let uε(x)≔u(x)−ε(x1+1){u}_{\varepsilon }\left(x):= u\left(x)-\varepsilon \left({x}_{1}+1)and define S0≔Σ∩{x1≥−1}andSε≔{uε≤0}.{\pmb{\mathscr{S}}}_{0}:= \Sigma \cap \left\{{x}_{1}\ge -1\right\}\hspace{1.0em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1.0em}{\pmb{\mathscr{S}}}_{\varepsilon }:= \left\{{u}_{\varepsilon }\le 0\right\}.Like before, for all ε≪1\varepsilon \ll 1, Sε⊂BD{\pmb{\mathscr{S}}}_{\varepsilon }\subset {\pmb{\mathscr{B}}}_{\pmb{\mathscr{D}}}for some D>0\pmb{\mathscr{D}}\gt 0. Here, however, as 0∈∂(dom(u))0\in \partial \left({\rm{dom}}\left(u)), we have that ∂u˜ε(S˜ε)⊃conv(Brn∪R+e1)withrn≔12n3/2.\partial {\tilde{u}}_{\varepsilon }\left({\tilde{\pmb{\mathscr{S}}}}_{\varepsilon })\supset {\rm{conv}}\left({\pmb{\mathscr{B}}}_{{r}_{n}}\cup {{\mathbb{R}}}^{+}{{\bf{e}}}_{1})\hspace{1.0em}\hspace{0.1em}\text{with}\hspace{0.1em}\hspace{0.33em}{r}_{n}:= \frac{1}{2{n}^{3\text{/}2}}.The function u˜ε{\tilde{u}}_{\varepsilon }is defined in an analogous fashion to how ϕ˜ε{\tilde{\phi }}_{\varepsilon }was defined in Case 2a (but replacing ϕ\phi by uu) and, again, S˜ε≔Aε(Sε){\tilde{\pmb{\mathscr{S}}}}_{\varepsilon }:= {\pmb{\mathscr{A}}}_{\varepsilon }\left({\pmb{\mathscr{S}}}_{\varepsilon })with Aε{\pmb{\mathscr{A}}}_{\varepsilon }denoting the John map associated to Sε{\pmb{\mathscr{S}}}_{\varepsilon }whose linear part is Lε{\pmb{\mathscr{L}}}_{\varepsilon }. In turn, arguing as we did in Case 2a, where again k∈Nk\in {\mathbb{N}}is such that 2n3/2≤2kdCnD2{n}^{3\text{/}2}\le {2}^{k}\frac{d}{{\pmb{\mathscr{C}}}_{n}\pmb{\mathscr{D}}}and Cμ{\pmb{\mathscr{C}}}_{\mu }is the doubling constant for μ\mu in B4Dn3/2{\pmb{\mathscr{B}}}_{4\pmb{\mathscr{D}}{n}^{3\text{/}2}}, but in the original variables, we deduce that νε(εLε(Kε))≤CμkνεεLεB6Dd{\nu }_{\varepsilon }(\varepsilon {\pmb{\mathscr{L}}}_{\varepsilon }\left({\pmb{\mathscr{K}}}_{\varepsilon }))\le {\pmb{\mathscr{C}}}_{\mu }^{k}{\nu }_{\varepsilon }\left(\varepsilon {\pmb{\mathscr{L}}}_{\varepsilon }\left({\pmb{\mathscr{B}}}_{\tfrac{6\pmb{\mathscr{D}}}{d}}\right)\right)for all ε≪1\varepsilon \ll 1(cf. (2.6)). Here, instead, νε≔(Id−εe1)#ν{\nu }_{\varepsilon }:= {\left({\rm{Id}}-\varepsilon {{\bf{e}}}_{1})}_{\#}\nu and Kε≔convBrn∪rnεe1.{\pmb{\mathscr{K}}}_{\varepsilon }:= {\rm{conv}}\left({\pmb{\mathscr{B}}}_{{r}_{n}}\cup \left\{\frac{{r}_{n}}{\varepsilon }{{\bf{e}}}_{1}\right\}\right).Now notice that ∣Lε(e1)∣=∣Aε(0)−Aε(−e1)∣≤2n3/2.| {\pmb{\mathscr{L}}}_{\varepsilon }\left({{\bf{e}}}_{1})| =| {\pmb{\mathscr{A}}}_{\varepsilon }\left(0)-{\pmb{\mathscr{A}}}_{\varepsilon }\left(-{{\bf{e}}}_{1})| \le 2{n}^{3\text{/}2}.Moreover, we claim there exists an N≫2n3/2>0\pmb{\mathscr{N}}\gg 2{n}^{3\text{/}2}\gt 0such that ‖εLε‖≤Nfor allε≪1.\Vert \varepsilon {\pmb{\mathscr{L}}}_{\varepsilon }\Vert \le \pmb{\mathscr{N}}\hspace{1.0em}\hspace{0.1em}\text{for all}\hspace{0.1em}\hspace{0.33em}\varepsilon \ll 1.Indeed, if not, then we can find a sequence of points zε∈Sε{z}_{\varepsilon }\in {\pmb{\mathscr{S}}}_{\varepsilon }and slopes pε∈∂uε(zε)∩span(S0)⊥{p}_{\varepsilon }\in \partial {u}_{\varepsilon }\left({z}_{\varepsilon })\cap {\rm{span}}{\left({\pmb{\mathscr{S}}}_{0})}^{\perp }such that ∣pε∣→∞| {p}_{\varepsilon }| \to \infty . In particular, in the limit, we find a point z0∈S0{z}_{0}\in {\pmb{\mathscr{S}}}_{0}such that ∂u(z0)∩span(S0)⊥\partial u\left({z}_{0})\cap {\rm{span}}{\left({\pmb{\mathscr{S}}}_{0})}^{\perp }contains a sequence of slopes {pj}j∈N{\left\{{p}_{j}\right\}}_{j\in {\mathbb{N}}}with ∣pj∣=j| {p}_{j}| =j. But as pj∈span(S0)⊥{p}_{j}\in {\rm{span}}{\left({\pmb{\mathscr{S}}}_{0})}^{\perp }, we see that pj⋅(x−z0)=pj⋅x=pj⋅(x−z){p}_{j}\cdot \left(x-{z}_{0})={p}_{j}\cdot x={p}_{j}\cdot \left(x-z)for any z∈S0z\in {\pmb{\mathscr{S}}}_{0}. Hence, pj∈∂u(z){p}_{j}\in \partial u\left(z)for all z∈S0z\in {\pmb{\mathscr{S}}}_{0}and j∈Nj\in {\mathbb{N}}. However, this is impossible; S0∩int(dom(u)){\pmb{\mathscr{S}}}_{0}\cap {\rm{int}}\left({\rm{dom}}\left(u))is nonempty, and on this set, uuis locally Lipschitz, proving the claim.Therefore, εLε(Kε)⊂BNandεLεB6Dd⊂B6DNd.\varepsilon {\pmb{\mathscr{L}}}_{\varepsilon }\left({\pmb{\mathscr{K}}}_{\varepsilon })\subset {\pmb{\mathscr{B}}}_{\pmb{\mathscr{N}}}\hspace{1.0em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1.0em}\varepsilon {\pmb{\mathscr{L}}}_{\varepsilon }\left({\pmb{\mathscr{B}}}_{\tfrac{6\pmb{\mathscr{D}}}{d}}\right)\subset {\pmb{\mathscr{B}}}_{\tfrac{6\pmb{\mathscr{D}}\pmb{\mathscr{N}}}{d}}.And so, arguing exactly like we did in Case 2a, we find that M≤CμkCνj+1,\pmb{\mathscr{M}}\le {\pmb{\mathscr{C}}}_{\mu }^{k}{\pmb{\mathscr{C}}}_{\nu }^{j+1},where Cν{\pmb{\mathscr{C}}}_{\nu }is the doubling constant for ν\nu in B6DN/d{\pmb{\mathscr{B}}}_{6\pmb{\mathscr{D}}\pmb{\mathscr{N}}\text{/}d}and M=M(ε)→∞\pmb{\mathscr{M}}=\pmb{\mathscr{M}}\left(\varepsilon )\to \infty as ε→0\varepsilon \to 0is the analogous count for this case’s Kε{\pmb{\mathscr{K}}}_{\varepsilon }(cf. (2.7)). But, again, this is impossible.Case 3. Σ\Sigma has an exposed point xˆ\hat{x}in Rn⧹X¯{{\mathbb{R}}}^{n}\setminus \overline{\pmb{\mathscr{X}}}. In this case, up to a translation, a dilation, a rotation, and subtracting ℓ\ell from uu, we can assume that xˆ=0,Σ⊂{x1≤0},ℓ≡0,andS0≔Σ∩{x1≥−1}⋐Rn⧹X¯.\hat{x}=0,\hspace{1.0em}\Sigma \subset \left\{{x}_{1}\le 0\right\},\hspace{1.0em}\ell \equiv 0,\hspace{1.0em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1.0em}{\pmb{\mathscr{S}}}_{0}:= \Sigma \cap \left\{{x}_{1}\ge -1\right\}\hspace{0.33em}\Subset \hspace{0.33em}{{\mathbb{R}}}^{n}\setminus \overline{\pmb{\mathscr{X}}}.Like before, let uε(x)≔u(x)−ε(x1+1){u}_{\varepsilon }\left(x):= u\left(x)-\varepsilon \left({x}_{1}+1)and define Sε≔{uε≤0}andνε≔(Id−εe1)#ν.{\pmb{\mathscr{S}}}_{\varepsilon }:= \left\{{u}_{\varepsilon }\le 0\right\}\hspace{1.0em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1.0em}{\nu }_{\varepsilon }:= {\left({\rm{Id}}-\varepsilon {{\bf{e}}}_{1})}_{\#}\nu .Again, Sε→S0{\pmb{\mathscr{S}}}_{\varepsilon }\to {\pmb{\mathscr{S}}}_{0}as ε→0\varepsilon \to 0, so diam(Sε)≤2diam(S0)andSε⋐Rn⧹X¯{\rm{diam}}\left({\pmb{\mathscr{S}}}_{\varepsilon })\le 2{\rm{diam}}\left({\pmb{\mathscr{S}}}_{0})\hspace{1.0em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1.0em}{\pmb{\mathscr{S}}}_{\varepsilon }\hspace{0.33em}\Subset \hspace{0.33em}{{\mathbb{R}}}^{n}\setminus \overline{\pmb{\mathscr{X}}}for all ε≪1\varepsilon \ll 1. For these small positive ε\varepsilon , then, 0=μ(Sε)=νε(∂uε(Sε)),0=\mu \left({\pmb{\mathscr{S}}}_{\varepsilon })={\nu }_{\varepsilon }\left(\partial {u}_{\varepsilon }\left({\pmb{\mathscr{S}}}_{\varepsilon })),where the second equality follows from the mass balance formula. (Recall that μ\mu vanishes on Rn⧹X¯{{\mathbb{R}}}^{n}\setminus \overline{\pmb{\mathscr{X}}}.) Moreover, as Y\pmb{\mathscr{Y}}is convex, ∂uε(Sε)⊂spt(νε)=Y¯−εe1\partial {u}_{\varepsilon }\left({\pmb{\mathscr{S}}}_{\varepsilon })\subset {\rm{spt}}\left({\nu }_{\varepsilon })=\overline{\pmb{\mathscr{Y}}}-\varepsilon {{\bf{e}}}_{1}(cf. (2.1)). Thus, any open subset of ∂uε(Sε)\partial {u}_{\varepsilon }\left({\pmb{\mathscr{S}}}_{\varepsilon })must be in the interior of the support of νε{\nu }_{\varepsilon }. In turn, considering the cone generated by ∂Sε\partial {\pmb{\mathscr{S}}}_{\varepsilon }over (0,uε(0))\left(0,{u}_{\varepsilon }\left(0)), for 0<ε≪10\lt \varepsilon \ll 1, we find that 0=νε(∂uε(Sε))≥νε(Brε)>0withrε≔∣uε(0)∣2diam(S0).0={\nu }_{\varepsilon }\left(\partial {u}_{\varepsilon }\left({\pmb{\mathscr{S}}}_{\varepsilon }))\ge {\nu }_{\varepsilon }\left({\pmb{\mathscr{B}}}_{{r}_{\varepsilon }})\gt 0\hspace{1.0em}\hspace{0.1em}\text{with}\hspace{0.1em}\hspace{0.33em}{r}_{\varepsilon }:= \frac{| {u}_{\varepsilon }\left(0)| }{2{\rm{diam}}\left({\pmb{\mathscr{S}}}_{0})}.(Again, for more details on this inclusion, see, e.g., [7, Theorem 2.8].) This is a contradiction and concludes the proof.□3Proof of Theorem 1.6Again, we replace uuby its lower-semicontinuous extension outside of X\pmb{\mathscr{X}}, exactly as we did at the beginning of Section 2. We split the proof into three parts.Part 1. uuis continuously differentiable inside X\pmb{\mathscr{X}}.We follow the argument used to prove [2, Corollary 1]. Assume for the sake of a contradiction that the result is false. Up to a translation, let 0∈X0\in \pmb{\mathscr{X}}be a point at which uuhas two distinct supporting planes. After a rotation, dilation, and subtracting off an affine function from uu, we may assume that u(x)≥max{x1,0},u(0)=0,andu(−se1)s→0ass→0.u\left(x)\ge \max \left\{{x}_{1},0\right\},\hspace{1.0em}u\left(0)=0,\hspace{1.0em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1.0em}\frac{u\left(-s{{\bf{e}}}_{1})}{s}\to 0\hspace{1em}\hspace{0.1em}\text{as}\hspace{0.1em}\hspace{0.33em}s\to 0.Now consider the function uσ{u}_{\sigma }defined by uσ(x)≔u(x)−τ(x1+2σ)withτ≔u(−σe1)σ.{u}_{\sigma }\left(x):= u\left(x)-\tau \left({x}_{1}+2\sigma )\hspace{1em}\hspace{0.1em}\text{with}\hspace{0.1em}\hspace{0.33em}\tau := \frac{u\left(-\sigma {{\bf{e}}}_{1})}{\sigma }.Note that τ→0\tau \to 0as σ→0\sigma \to 0. If Sσ≔{uσ≤0},{\pmb{\mathscr{S}}}_{\sigma }:= \left\{{u}_{\sigma }\le 0\right\},then, by the strict convexity of uuprovided by Theorem 1.2, we see that Sσ⊂BD⋐X∩int(dom(u)){\pmb{\mathscr{S}}}_{\sigma }\subset {\pmb{\mathscr{B}}}_{\pmb{\mathscr{D}}}\hspace{0.33em}\Subset \hspace{0.33em}\pmb{\mathscr{X}}\cap {\rm{int}}\left({\rm{dom}}\left(u))for some D>0\pmb{\mathscr{D}}\gt 0and for all σ≪1\sigma \ll 1; also, for these small positive σ\sigma , ∂uσ(Sσ)⊂BR∩Y\partial {u}_{\sigma }\left({\pmb{\mathscr{S}}}_{\sigma })\subset {\pmb{\mathscr{B}}}_{\pmb{\mathscr{R}}}\cap \pmb{\mathscr{Y}}for some R>0\pmb{\mathscr{R}}\gt 0. Moreover, if Π−a≔{x1=−a}{\Pi }_{-a}:= \left\{{x}_{1}=-a\right\}and Πb≔{x1=b}{\Pi }_{b}:= \left\{{x}_{1}=b\right\}denote the two parallel planes that tangentially sandwich Sσ{\pmb{\mathscr{S}}}_{\sigma }, we see that a>σandb<2τσ1−τ,a\gt \sigma \hspace{1.0em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1.0em}b\lt \frac{2\tau \sigma }{1-\tau },provided σ>0\sigma \gt 0is small enough to guarantee that τ<1\tau \lt 1. Furthermore, maxSσ∣uσ∣=∣uσ(0)∣=2τσ,\mathop{\max }\limits_{{\pmb{\mathscr{S}}}_{\sigma }}| {u}_{\sigma }| =| {u}_{\sigma }\left(0)| =2\tau \sigma ,and dist(Πb,Π0)dist(Π−a,Πb)=ba+b≤ba≤2τ1−τ→0asσ→0,\frac{{\rm{dist}}\left({\Pi }_{b},{\Pi }_{0})}{{\rm{dist}}\left({\Pi }_{-a},{\Pi }_{b})}=\frac{b}{a+b}\le \frac{b}{a}\le \frac{2\tau }{1-\tau }\to 0\hspace{1.0em}\hspace{0.1em}\text{as}\hspace{0.1em}\hspace{0.33em}\sigma \to 0,where Π0≔{x1=0}{\Pi }_{0}:= \left\{{x}_{1}=0\right\}.Now, set u˜σ(x)≔uσ(Aσ−1x)2τσandS˜σ≔Aσ(Sσ),{\tilde{u}}_{\sigma }\left(x):= \frac{{u}_{\sigma }\left({\pmb{\mathscr{A}}}_{\sigma }^{-1}x)}{2\tau \sigma }\hspace{1.0em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1.0em}{\tilde{\pmb{\mathscr{S}}}}_{\sigma }:= {\pmb{\mathscr{A}}}_{\sigma }\left({\pmb{\mathscr{S}}}_{\sigma }),where Aσ{\pmb{\mathscr{A}}}_{\sigma }is the John map that normalizes Sσ{\pmb{\mathscr{S}}}_{\sigma }. Arguing as we did in the proof of Theorem 1.2, we find the same contradiction as we did in Case 2 when τ\tau sufficiently small; the only difference is that we consider a slightly different chain of inequalities: ν˜σ(Kσ)≤ν˜σ(∂u˜σ(S˜σ))=μ˜σ(S˜σ)≤Cμμ˜σ12S˜σ=Cμν˜σ∂u˜σ12S˜σ≤Cμν˜σ(B1rn){\tilde{\nu }}_{\sigma }\left({\pmb{\mathscr{K}}}_{\sigma })\le {\tilde{\nu }}_{\sigma }(\partial {\tilde{u}}_{\sigma }\left({\tilde{\pmb{\mathscr{S}}}}_{\sigma }))={\tilde{\mu }}_{\sigma }\left({\tilde{\pmb{\mathscr{S}}}}_{\sigma })\le {\pmb{\mathscr{C}}}_{\mu }{\tilde{\mu }}_{\sigma }\left(\frac{1}{2}{\tilde{\pmb{\mathscr{S}}}}_{\sigma }\right)={\pmb{\mathscr{C}}}_{\mu }{\tilde{\nu }}_{\sigma }\left(\partial {\tilde{u}}_{\sigma }\left(\frac{1}{2}{\tilde{\pmb{\mathscr{S}}}}_{\sigma }\right)\right)\le {\pmb{\mathscr{C}}}_{\mu }{\tilde{\nu }}_{\sigma }\left({\pmb{\mathscr{B}}}_{\tfrac{1}{{r}_{n}}})(cf. (2.6)), where μ˜σ{\tilde{\mu }}_{\sigma }and ν˜σ{\tilde{\nu }}_{\sigma }are defined so that (∇u˜σ)#μ˜σ=ν˜σ{\left(\nabla {\tilde{u}}_{\sigma })}_{\#}{\tilde{\mu }}_{\sigma }={\tilde{\nu }}_{\sigma }and Kσ≔convBrn∪rn+n(1−τ)2τe1withrn≔12n3/2.{\pmb{\mathscr{K}}}_{\sigma }:= {\rm{conv}}\left({\pmb{\mathscr{B}}}_{{r}_{n}}\cup \left\{\phantom{\rule[-1.25em]{}{0ex}},\frac{{r}_{n}+n\left(1-\tau )}{2\tau }{{\bf{e}}}_{1}\right\}\right)\hspace{1.0em}\hspace{0.1em}\text{with}\hspace{0.1em}\hspace{0.33em}{r}_{n}:= \frac{1}{2{n}^{3\text{/}2}}.This proves that uuis differentiable.By [7, Lemma A.24], for example, we know that differentiable convex functions are continuously differentiable. So we conclude that uuis continuously differentiable in X\pmb{\mathscr{X}}.Part 2. ∇u(X)=Y\nabla u\left(\pmb{\mathscr{X}})=\pmb{\mathscr{Y}}when X\pmb{\mathscr{X}}is convex.Because ∇u\nabla uis continuous in X\pmb{\mathscr{X}}, its image Y′≔∇u(X)\pmb{\mathscr{Y}}^{\prime} := \nabla u\left(\pmb{\mathscr{X}})is an open set of full ν\nu -measure contained inside Y\pmb{\mathscr{Y}}. Also, as the assumptions on μ\mu and ν\nu are symmetric, the optimal transport map ∇v\nabla vfrom ν\nu to μ\mu is continuous, and X′≔∇v(Y)\pmb{\mathscr{X}}^{\prime} := \nabla v\left(\pmb{\mathscr{Y}})is an open set of full μ\mu -measure contained inside X\pmb{\mathscr{X}}. Hence, by recalling that ∇u\nabla uand ∇v\nabla vare inverses of each other [8, Corollary 2.5.13], we conclude that X′=X\pmb{\mathscr{X}}^{\prime} =\pmb{\mathscr{X}}and Y′=Y\pmb{\mathscr{Y}}^{\prime} =\pmb{\mathscr{Y}}, as desired.Part 3. ∇u\nabla uis locally Hölder continuous inside X\pmb{\mathscr{X}}.Thanks to the strict convexity and C1{\pmb{\mathscr{C}}}^{1}regularity of uu, we can localize the arguments of the proof of [11, Theorem 1.1] to obtain the local Hölder continuity of uuinside X\pmb{\mathscr{X}}.More precisely, if u∗{u}^{\ast }denotes the Legendre transform of uu(2.2), as in [11], one can show that u∗{u}^{\ast }satisfies a weak form of Alexandrov’s maximum principle [11, Lemma 3.2], from which one deduces the engulfing property for the sections of u∗{u}^{\ast }[11, Lemma 3.3]. Iteratively applying this engulfing property, one obtains a polynomial strict convexity bound for u∗{u}^{\ast }. This bound implies the local Hölder continuity of uuinside X\pmb{\mathscr{X}}[11, Proof of Theorem 1.1]. We leave the details of this adaptation to the interested reader. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Advanced Nonlinear Studies de Gruyter

Regularity properties of monotone measure-preserving maps

Figalli, Alessio; Jhaveri, Yash
Advanced Nonlinear Studies , Volume 23 (1): 1 – Jan 1, 2023
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de Gruyter
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© 2023 the author(s), published by De Gruyter
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1536-1365
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2169-0375
DOI
10.1515/ans-2022-0057
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Abstract

1IntroductionGiven a pair of atom-free Borel probability measures μ\mu and ν\nu on R{\mathbb{R}}, the monotone rearrangement theorem asserts that the function y=y(x)y=y\left(x)defined implicitly by ∫−∞xdμ=∫−∞y(x)dν\underset{-\infty }{\overset{x}{\int }}{\rm{d}}\mu =\underset{-\infty }{\overset{y\left(x)}{\int }}{\rm{d}}\nu is measure preserving, i.e., μ(y−1(E))=ν(E)for any Borel setE⊂R.\mu ({y}^{-1}\left(\pmb{\mathscr{E}}))=\nu \left(\pmb{\mathscr{E}})\hspace{1.0em}\hspace{0.1em}\text{for any Borel set}\hspace{0.1em}\hspace{0.33em}\pmb{\mathscr{E}}\subset {\mathbb{R}}.In particular, yyis unique μ\mu almost everywhere and can be made to be monotone, or, equivalently, the derivative of a convex function.When R{\mathbb{R}}is replaced by Rn{{\mathbb{R}}}^{n}, however, before Brenier’s discovery [1], a proper generalization of the monotone rearrangement theorem was myth, and only after the work of McCann, in [12], was the myth made real. Precisely, he proved that if μ\mu vanishes on every Lipschitz (n−1)\left(n-1)-dimensional surface,McCann actually assumes that μ\mu vanishes on all (Borel) sets of Hausdorff dimension n−1n-1. This guarantees that the set of nondifferentiability points of a convex function are μ\mu -negligible. However, this assumption can be weakened. Since convex functions are differentiable outside of a countable union of Lipschitz hypersurfaces [14], McCann’s theorem holds assuming that μ\mu vanishes on every Lipschitz (n−1)\left(n-1)-dimensional surface.then a convex potential u:Rn→R∪{+∞}u:{{\mathbb{R}}}^{n}\to {\mathbb{R}}\cup \left\{+\infty \right\}exists whose gradient map ∇u=∇u(x)\nabla u=\nabla u\left(x)is unique μ\mu almost everywhere and pushes μ\mu forward to ν\nu , i.e., μ((∇u)−1(E))=ν(E)for any Borel setE⊂Rn.\mu \left({\left(\nabla u)}^{-1}\left(\pmb{\mathscr{E}}))=\nu \left(\pmb{\mathscr{E}})\hspace{1.0em}\hspace{0.1em}\text{for any Borel set}\hspace{0.1em}\hspace{0.33em}\pmb{\mathscr{E}}\subset {{\mathbb{R}}}^{n}.(Brenier’s theorem guaranteed the same conclusion as McCann’s theorem, but under some restrictive technical conditions on μ\mu and ν\nu .)The first general regularity result on these Brenier-McCann maps was proved by Caffarelli in [4]: provided that μ\mu and ν\nu are absolutely continuous with respect to nn-dimensional Lebesgue measure Ln{{\mathscr{L}}}^{n}, their respective densities ffand ggvanish outside of and are bounded away from zero and infinity on open bounded sets X\pmb{\mathscr{X}}and Y\pmb{\mathscr{Y}}, respectively, and Y\pmb{\mathscr{Y}}is convex, he showed that uuis strictly convex in X\pmb{\mathscr{X}}(see, e.g., the proof of [7, Theorem 4.6.2]). This result opened the door to the development of a regularity theory for mappings with convex potentials based on the regularity theory for strictly convex solutions to the Monge-Ampère equation [2]; indeed, (∇u)#f=gis formally, at least, equivalent todetD2u=fg(∇u).{\left(\nabla u)}_{\#}f=g\hspace{0.33em}\hspace{0.1em}\text{is formally, at least, equivalent to}\hspace{0.1em}\hspace{0.33em}\det {\pmb{\mathscr{D}}}^{2}u=\frac{f}{g\left(\nabla u)}.Unfortunately, Caffarelli’s boundedness assumptions on the domains X\pmb{\mathscr{X}}and Y\pmb{\mathscr{Y}}are restrictive, since many probability densities, especially those found in applications, are supported on all of Rn{{\mathbb{R}}}^{n}: Gaussian densities, for example. Motivated by this, in [5], Cordero-Erausquin and Figalli showed that, in several situations of interest, one can ensure the regularity of monotone measure-preserving maps even if the measures under consideration have unbounded supports. However, missing from their collection is the situation where Y\pmb{\mathscr{Y}}is an arbitrary convex domain. Lifting this restriction is a main goal of this article.1.1ResultsOur main theorem is an extension of Caffarelli’s theorem, on the strict convexity of uu, in two ways. First, we allow X\pmb{\mathscr{X}}and Y\pmb{\mathscr{Y}}to be unbounded. Second, we permit X\pmb{\mathscr{X}}and Y\pmb{\mathscr{Y}}to carrying certain invariant measures that we call locally doubling measures (qualitatively, our notion replaces balls in the classical notion of a doubling measure with ellipsoids, in order to account for the affine invariance of our setting).Definition 1.1A nonnegative measure λ\lambda is locally doubling (on ellipsoids) if the following holds: for every ball B\pmb{\mathscr{B}}, there is a constant C≥1\pmb{\mathscr{C}}\ge 1such that λ(ℰ)≤Cλ12ℰ\lambda \left({\mathcal{ {\mathcal E} }})\le \pmb{\mathscr{C}}\lambda \left(\frac{1}{2}{\mathcal{ {\mathcal E} }}\right)for all ellipsoids ℰ⊂B{\mathcal{ {\mathcal E} }}\subset \pmb{\mathscr{B}}with center (of mass) in spt(λ){\rm{spt}}\left(\lambda ). Here, 12ℰ\frac{1}{2}{\mathcal{ {\mathcal E} }}is the dilation of ℰ{\mathcal{ {\mathcal E} }}with respect to its center by 1/2.This notion of doubling was introduced by Jhaveri and Savin in [11].This family of measures is strictly larger than the family of measures locally comparable to Lebesgue measure on their supports. (See [11] for examples of locally doubling measures not comparable to Lebesgue measure on their supports.)That said, the first consideration of measures with a “doubling-like” property in the world of solutions to Monge-Ampère equations can be traced back to the work of Jerison [9] and then Caffarelli [3]. In particular, in [3], Caffarelli showed that Alexandrov solutions to detD2v=ρ,\det {\pmb{\mathscr{D}}}^{2}v=\rho ,where the measure ρ\rho is doubling on a specific collection of convex sets called sectionsThese are sets of the form {v≤ℓ}\left\{v\le \ell \right\}for any affine function ℓ\ell ., share the same geometric properties as Alexandrov solutions to Monge-Ampère equations with right-hand sides comparable to Lebesgue measure [2].We now state our main theorem.Theorem 1.2Let μ\mu and ν\nu be two locally doubling probability measures on Rn{{\mathbb{R}}}^{n}that vanish on Lipschitz (n−1)\left(n-1)-dimensional surfaces and are concentrated on two open sets X\pmb{\mathscr{X}}and Y\pmb{\mathscr{Y}}, respectively, and suppose that Y\pmb{\mathscr{Y}}is convex. Then any convex potential u associated to the Brenier-McCann map pushing μ\mu forward to ν\nu is strictly convex in X\pmb{\mathscr{X}}.Remark 1.3It is well known that Y\pmb{\mathscr{Y}}needs to be convex. When Y\pmb{\mathscr{Y}}is not convex, uucan fail to be strictly convex and ∇u\nabla ucan behave rather poorly. If we consider Pogorelov’s counterexample to the strict convexity of solutions to the Monge-Ampère equation in three dimensions, we see that Y\pmb{\mathscr{Y}}needs to be convex in order to guarantee the strict convexity of potentials of Brenier-McCann maps. In particular, let u(x′,x3)=∣x′∣4/3(1+x32)u\left(x^{\prime} ,{x}_{3})=| x^{\prime} {| }^{4\text{/}3}\left(1+{x}_{3}^{2}), let Qr={∣xi∣<r/2fori=1,2,3}{\pmb{\mathscr{Q}}}_{r}=\left\{| {x}_{i}| \lt r\hspace{0.1em}\text{/}\hspace{0.1em}2\hspace{0.33em}\hspace{0.1em}\text{for}\hspace{0.1em}\hspace{0.33em}i=1,2,3\right\}be the cube with side length r>0r\gt 0and centered at the origin in R3{{\mathbb{R}}}^{3}, and let Y=∇u(Qr)\pmb{\mathscr{Y}}=\nabla u\left({\pmb{\mathscr{Q}}}_{r}). If r≪1r\ll 1, then f=detD2uf=\det {\pmb{\mathscr{D}}}^{2}uis analytic and positive in Qr{\pmb{\mathscr{Q}}}_{r}, and ∇u\nabla uis the Brenier-McCann map pushing forward μ=(f/‖f‖L1(Qr))L3∣__Qr\mu =(f\hspace{0.1em}\text{/}\hspace{0.1em}\Vert f{\Vert }_{{\pmb{\mathscr{L}}}^{1}\left({\pmb{\mathscr{Q}}}_{r})}){{\mathscr{L}}}^{3}| \hspace{-0.3em}\hspace{0.1em}\text{\_\_}\hspace{0.1em}{\pmb{\mathscr{Q}}}_{r}to ν=(1/L3(Y))L3∣__Y\nu =\left(1\hspace{0.1em}\text{/}\hspace{0.1em}{{\mathscr{L}}}^{3}\left(\pmb{\mathscr{Y}})){{\mathscr{L}}}^{3}| \hspace{-0.3em}\hspace{0.1em}\text{\_\_}\hspace{0.1em}\pmb{\mathscr{Y}}. The set Y\pmb{\mathscr{Y}}is open but not convex, and u=0u=0along {x′=0}\left\{x^{\prime} =0\right\}. Moreover, as demonstrated in, for instance, [4,10], ∇u\nabla ueasily fails to be continuous when Y\pmb{\mathscr{Y}}is not convex.Remark 1.4While the target measure ν\nu need not vanishes on all Lipschitz (n−1)\left(n-1)-dimensional surfaces in order to invoke McCann’s theorem (which asks this only of the source measure μ\mu ), it must in order to ensure that our main theorem holds. If μ=L2∣__Q1\mu ={{\mathscr{L}}}^{2}| \hspace{-0.3em}\hspace{0.1em}\text{\_\_}\hspace{0.1em}{\pmb{\mathscr{Q}}}_{1}is the uniform measure on Q1{\pmb{\mathscr{Q}}}_{1}the unit cube centered at the origin in R2{{\mathbb{R}}}^{2}and ν=H1∣__Q1∩{x2=0}\nu ={{\mathscr{H}}}^{1}| \hspace{-0.3em}\hspace{0.1em}\text{\_\_}\hspace{0.1em}{\pmb{\mathscr{Q}}}_{1}\cap \left\{{x}_{2}=0\right\}is the 1-dimensional Hausdorff measure restricted to the central horizontal axis of Q1{\pmb{\mathscr{Q}}}_{1}, then the Brenier-McCann map pushing μ\mu forward to ν\nu is the projection map (x1,x2)↦x1\left({x}_{1},{x}_{2})\mapsto {x}_{1}. Up to a constant, this map’s convex potential is 12x12\frac{1}{2}{x}_{1}^{2}, which is not strictly convex. With respect our proof of Theorem 1.2, asking this of both μ\mu and ν\nu guarantees the validity of the mass balance formula in Lemma 2.1 (see also Remark 2.2), our main tool.Remark 1.5A simple case to which Theorem 1.2 applies, but the corresponding results in [4, 5] do not, is when μ=gL3∣__{∣x1∣<1}×R2\mu =g{{\mathscr{L}}}^{3}| \hspace{-0.3em}\hspace{0.1em}\text{\_\_}\hspace{0.1em}\left\{| {x}_{1}| \lt 1\right\}\times {{\mathbb{R}}}^{2}and ν=gL3∣__R2×{∣x3∣<1}\nu =g{{\mathscr{L}}}^{3}| \hspace{-0.3em}\hspace{0.1em}\text{\_\_}\hspace{0.1em}{{\mathbb{R}}}^{2}\times \left\{| {x}_{3}| \lt 1\right\}, and ggis the standard Gaussian density on R3{{\mathbb{R}}}^{3}appropriately normalized to make μ\mu and ν\nu probability measures.With our main theorem in hand, our second and third theorems further extend the known regularity theory for monotone measure-preserving maps, completing the story started by Cordero-Erausquin and Figalli in [5] on monotone transports between unbounded domains.Theorem 1.6Let μ\mu and ν\nu be two locally doubling probability measures on Rn{{\mathbb{R}}}^{n}that vanish on Lipschitz (n−1)\left(n-1)-dimensional surfaces and are concentrated on two open sets X\pmb{\mathscr{X}}and Y\pmb{\mathscr{Y}}, respectively, and suppose that Y\pmb{\mathscr{Y}}is convex. Then the Brenier-McCann map ∇u\nabla upushing μ\mu forward to ν\nu is a homeomorphism from X\pmb{\mathscr{X}}onto a full measure subset of Y\pmb{\mathscr{Y}}. Moreover, for every A⋐X\pmb{\mathscr{A}}\hspace{0.33em}\Subset \hspace{0.33em}\pmb{\mathscr{X}}, a constant α>0\alpha \gt 0exists such that ∇u∈C0,α(A)\nabla u\in {\pmb{\mathscr{C}}}^{0,\alpha }\left(\pmb{\mathscr{A}}). Furthermore, ∇u(X)=Y\nabla u\left(\pmb{\mathscr{X}})=\pmb{\mathscr{Y}}whenever X\pmb{\mathscr{X}}is convex.Theorem 1.7Let f and g be two functions on Rn{{\mathbb{R}}}^{n}that define locally doubling probability measures concentrated on two open sets X\pmb{\mathscr{X}}and Y\pmb{\mathscr{Y}}, respectively, and suppose that Y\pmb{\mathscr{Y}}is convex. Assume that f and g are bounded away from zero and infinity on compact subsets of X\pmb{\mathscr{X}}and Y\pmb{\mathscr{Y}}, respectively. Then for every E⋐X\pmb{\mathscr{E}}\hspace{0.33em}\Subset \hspace{0.33em}\pmb{\mathscr{X}}, a constant ε>0\varepsilon \gt 0exists such that any convex potential u associated to the Brenier-McCann map pushing f forward to g is W2,1+ε(E){\pmb{\mathscr{W}}}^{2,1+\varepsilon }\left(\pmb{\mathscr{E}}). Also, ∇u\nabla uis locally a Ck+1,β{\pmb{\mathscr{C}}}^{k+1,\beta }-diffeomorphism from X\pmb{\mathscr{X}}onto its image provided f and g are locally Ck,β{\pmb{\mathscr{C}}}^{k,\beta }in X\pmb{\mathscr{X}}and Y\pmb{\mathscr{Y}}, respectively.Remark 1.8We note that the proof of the Theorem 1.7, given the strict convexity of uu(provided by Theorem 1.2), is classical. Indeed, it suffices to localize classical regularity results for the Monge-Ampère equation. We refer the reader to [7, Section 4.6.1] for more details.1.2StructureThis remainder of this article is structured as follows.In Section 2, we prove Theorem 1.2. Our proof is self-contained apart from some facts in convex analysis; we provide explicit references to these used but unproved facts. We remark that our proof is inspired by the proof of the Alexandrov maximum principle in [11] (and, of course, Caffarelli’s original proof of the strict convexity of potential functions of optimal transports/solutions to Monge-Ampère equations). If the reader is familiar with [5] or [4], then they might consider directing their attention to Case 2. Case 2b is completely novel. Case 2a illustrates our argument in the setting of [4], which builds on the work of [2] and is the foundation for Case 2b.Section 3 is dedicated to the proof of Theorem 1.6. Our proof here is similarly self-contained (and an adaptation of Caffarelli’s argument of the same result in [2], but, of course, using the line of reasoning developed to prove Theorem 1.2). The Hölder regularity of ∇u\nabla uis a consequence of appropriately localizing the arguments of [11].2Proof of Theorem 1.2Before we begin, it will be convenient to replace the potential uuby the following lower-semicontinuous extension of uuoutside of X\pmb{\mathscr{X}}:Here, ∂u(z)\partial u\left(z)is called the subdifferential of uuat zzand is defined as follows: ∂u(z)≔{p∈Rn:u(x)≥u(z)+p⋅(x−z)for allx∈X}.\partial u\left(z):= \left\{p\in {{\mathbb{R}}}^{n}:u\left(x)\ge u\left(z)+p\cdot \left(x-z)\hspace{0.4em}\text{for all}\hspace{0.4em}\hspace{0.1em}x\in \pmb{\mathscr{X}}\text{}\right\}.Moreover, for a set E⊂Rn\pmb{\mathscr{E}}\subset {{\mathbb{R}}}^{n}, we define ∂u(E)≔∪z∈E∂u(z)\partial u\left(\pmb{\mathscr{E}}):= {\cup }_{z\in \pmb{\mathscr{E}}}\partial u\left(z).u̲(x)≔supz∈Xp∈∂u(z){u(z)+p⋅(x−z)}.\underline{u}\left(x):= \mathop{\sup }\limits_{\begin{array}{c}z\in \pmb{\mathscr{X}}\\ p\in \partial u\left(z)\end{array}}\left\{u\left(z)+p\cdot \left(x-z)\right\}.Observe that u̲∣X=u∣X\underline{u}{| }_{\pmb{\mathscr{X}}}=u{| }_{\pmb{\mathscr{X}}}. For notational simplicity, we shall not distinguish u̲\underline{u}from uu; so when we write uuin what follows, we mean u̲\underline{u}.We shall denote the domain of uuby dom(u){\rm{dom}}\left(u), namely, dom(u)≔{u<+∞}{\rm{dom}}\left(u):= \left\{u\lt +\infty \right\}. Note that dom(u){\rm{dom}}\left(u)is convex. We recall that convex functions are locally Lipschitz inside their domain [7, Appendix A.4]. Furthermore, we shall denote the convex hull of a set A\pmb{\mathscr{A}}by conv(A){\rm{conv}}\left(\pmb{\mathscr{A}}).Let ℓ\ell define a supporting plane to the graph of uuat a point in X\pmb{\mathscr{X}}. Precisely, ℓ(x)=u(z)+p⋅(x−z)for some(z,p)∈X×Rn\ell \left(x)=u\left(z)+p\cdot \left(x-z)\hspace{1.0em}\hspace{0.1em}\text{for some}\hspace{0.1em}\hspace{0.33em}\left(z,p)\in \pmb{\mathscr{X}}\times {{\mathbb{R}}}^{n}and ℓ≤u\ell \le u. Note that Σ≔{u=ℓ}={u≤ℓ}\Sigma := \left\{u=\ell \right\}=\left\{u\le \ell \right\}is closed, as uuis lower-semicontinuous, Σ,X⊂dom(u),\Sigma ,\pmb{\mathscr{X}}\subset {\rm{dom}}\left(u),and, because Y\pmb{\mathscr{Y}}is convex, (2.1)∂u(Rn)⊂∂u(X)¯⊂Y¯andLn(Y⧹∂u(X))=0.\partial u\left({{\mathbb{R}}}^{n})\subset \overline{\partial u\left(\pmb{\mathscr{X}})}\subset \overline{\pmb{\mathscr{Y}}}\hspace{1.0em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1.0em}{{\mathscr{L}}}^{n}\left(\pmb{\mathscr{Y}}\setminus \partial u\left(\pmb{\mathscr{X}}))=0.(A proof of (2.1) can be found in [5].)Recall that an exposed point xˆ\hat{x}of Σ⊂Rn\Sigma \subset {{\mathbb{R}}}^{n}is one for which there exists a hyperplane Π⊂Rn\Pi \subset {{\mathbb{R}}}^{n}tangent to Σ\Sigma at xˆ\hat{x}such that Π∩Σ={xˆ}\Pi \cap \Sigma =\left\{\hat{x}\right\}. Also, remember that optimal/monotone transports balance mass, in the following way.Lemma 2.1(Mass balance formula) Let u:Rn→R∪{+∞}u:{{\mathbb{R}}}^{n}\to {\mathbb{R}}\cup \left\{+\infty \right\}be convex and such that (∇u)#μ=ν{\left(\nabla u)}_{\#}\mu =\nu where μ\mu and ν\nu are two Borel measures that vanish on all Lipschitz (n−1)\left(n-1)-dimensional surfaces. Then for all Borel sets E⊂Rn\pmb{\mathscr{E}}\subset {{\mathbb{R}}}^{n}, μ(E)=ν(∂u(E)).\mu \left(\pmb{\mathscr{E}})=\nu \left(\partial u\left(\pmb{\mathscr{E}})).Remark 2.2The mass balance formula was originally proved for measures that are absolutely continuous with respect to Lebesgue measure [13, Lemma 4.6]. However, with respect to absolute continuity, the proof only relies on the measures in question not giving mass to the set of nondifferentiable points of a convex function. As observed in Section 1, such points are contained in a countable union of Lipschitz (n−1)\left(n-1)-dimensional surfaces. So the set of nondifferentiable points of a convex function is negligible both for μ\mu and ν\nu under our assumption.Finally, recall that if a nonnegative measure is locally doubling (on ellipsoids), then it is locally doubling on all bounded convex domains [11, Corollary 2.5]. After this preliminary discussion, we can now prove our main theorem.Proof of Theorem 1.2Proving that uuis strictly convex in X\pmb{\mathscr{X}}corresponds to proving that for any supporting plane ℓ\ell to (the graph of) uuat a point in X\pmb{\mathscr{X}}, the set Σ={u=ℓ}\Sigma =\left\{u=\ell \right\}is a singleton. Assuming that Σ\Sigma is not a singleton, we will show that Σ\Sigma both has and does not have exposed points, which cannot be; thus, Σ\Sigma is a singleton, as desired.Case 1. Σ\Sigma has no exposed points. If Σ\Sigma has no exposed points, then Σ⊃Re\Sigma \supset {\mathbb{R}}{\bf{e}}for some unit vector e{\bf{e}}. In turn, ∂u(Rn)⊂e⊥\partial u\left({{\mathbb{R}}}^{n})\subset {{\bf{e}}}^{\perp }[7, Lemma A.25]. But this is impossible given (2.1): 0<Ln(∂u(X)∩Y)≤Ln(∂u(Rn)∩Y)≤Ln(e⊥∩Y)=0.0\lt {{\mathscr{L}}}^{n}\left(\partial u\left(\pmb{\mathscr{X}})\cap \pmb{\mathscr{Y}})\le {{\mathscr{L}}}^{n}\left(\partial u\left({{\mathbb{R}}}^{n})\cap \pmb{\mathscr{Y}})\le {{\mathscr{L}}}^{n}\left({{\bf{e}}}^{\perp }\cap \pmb{\mathscr{Y}})=0.Case 2. Σ\Sigma has an exposed point xˆ\hat{x}in X¯\overline{\pmb{\mathscr{X}}}. Up to a translation and a rotation, we can assume that xˆ=0∈X¯,Σ⊂{x1≤0},andΣ∩{x1=0}={0}.\hat{x}=0\in \overline{\pmb{\mathscr{X}}},\hspace{1.0em}\Sigma \subset \left\{{x}_{1}\le 0\right\},\hspace{1.0em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1.0em}\Sigma \cap \left\{{x}_{1}=0\right\}=\left\{0\right\}.Since X\pmb{\mathscr{X}}is open and Σ∩X\Sigma \cap \pmb{\mathscr{X}}is nonempty by construction, there is a point xint∈Σ∩X{x}_{int}\in \Sigma \cap \pmb{\mathscr{X}}and a ball centered at this point completely contained in X\pmb{\mathscr{X}}. Thus, up to a shearing transformation x↦x−ηx1x\mapsto x-\eta {x}_{1}with η⋅e1=0\eta \cdot {{\bf{e}}}_{1}=0, and a dilation, we may assume that xint=−e1andBd(−e1)⋐X{x}_{int}=-{{\bf{e}}}_{1}\hspace{1.0em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1.0em}{\pmb{\mathscr{B}}}_{d}\left(-{{\bf{e}}}_{1})\hspace{0.33em}\Subset \hspace{0.33em}\pmb{\mathscr{X}}for some d>0d\gt 0. Finally, up to subtracting ℓ\ell from uu, we can assume that ℓ≡0.\ell \equiv 0.Case 2a: 0∈int(dom(u))∩X¯0\in {\rm{int}}\left({\rm{dom}}\left(u))\cap \overline{\pmb{\mathscr{X}}}. As our exposed point 0 and all of the points in Bd(−e1)¯\overline{{\pmb{\mathscr{B}}}_{d}\left(-{{\bf{e}}}_{1})}belong to int(dom(u)){\rm{int}}\left({\rm{dom}}\left(u)), which is convex (and, by definition, open), the convex hull of the union of Bd(−e1)¯\overline{{\pmb{\mathscr{B}}}_{d}\left(-{{\bf{e}}}_{1})}and {0}\left\{0\right\}is contained in int(dom(u)){\rm{int}}\left({\rm{dom}}\left(u)). So there exists an open, bounded set U⋐int(dom(u))\pmb{\mathscr{U}}\hspace{0.33em}\Subset \hspace{0.33em}{\rm{int}}\left({\rm{dom}}\left(u))containing conv(Bd(−e1)¯∪{0}){\rm{conv}}\left(\overline{{\pmb{\mathscr{B}}}_{d}\left(-{{\bf{e}}}_{1})}\cup \left\{0\right\}). Moreover, we know that ∂u(U)⊂conv(∇u(U)¯)≕ϒ⊂BR∩Y¯\partial u\left(\pmb{\mathscr{U}})\subset {\rm{conv}}\left(\overline{\nabla u\left(\pmb{\mathscr{U}})})\hspace{0.33em}=: \hspace{0.33em}\Upsilon \subset {\pmb{\mathscr{B}}}_{\pmb{\mathscr{R}}}\cap \overline{\pmb{\mathscr{Y}}}for some R>0\pmb{\mathscr{R}}\gt 0[7, Lemma A.22]. Let u∗{u}^{\ast }be the Legendre transform of uu, namely, (2.2)u∗(q)≔supx∈Rn{q⋅x−u(x)}{u}^{\ast }\left(q):= \mathop{\sup }\limits_{x\in {{\mathbb{R}}}^{n}}\left\{q\cdot x-u\left(x)\right\}and define Ω≔∂u∗(ϒ)⊃U.\Omega := \partial {u}^{\ast }\left(\Upsilon )\supset \pmb{\mathscr{U}}.Recalling that ∂u\partial uand ∂u∗\partial {u}^{\ast }are inverses of each other [7, Section A.4.2], we deduce that (∇u)#ρ=γ{\left(\nabla u)}_{\#}\rho =\gamma , where ρ≔μ∣__Ωandγ≔ν∣__ϒ.\rho := \mu | \hspace{-0.3em}\hspace{0.1em}\text{\_\_}\hspace{0.1em}\Omega \hspace{1.0em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1.0em}\gamma := \nu | \hspace{-0.3em}\hspace{0.1em}\text{\_\_}\hspace{0.1em}\Upsilon .In particular, if we let ϕ:Rn→R∪{+∞}\phi :{{\mathbb{R}}}^{n}\to {\mathbb{R}}\cup \left\{+\infty \right\}be defined by ϕ(x)≔supz∈Ωp∈∂u(z){u(z)+p⋅(x−z)},\phi \left(x):= \mathop{\sup }\limits_{\begin{array}{c}z\in \Omega \\ p\in \partial u\left(z)\end{array}}\left\{u\left(z)+p\cdot \left(x-z)\right\},then, by construction, ϕ\phi and uuagree on Ω\Omega , ϕ\phi is (globally) Lipschitz, ∂ϕ(Rn)=∂ϕ(Ω)=ϒ,\partial \phi \left({{\mathbb{R}}}^{n})=\partial \phi \left(\Omega )=\Upsilon ,and 0∈Ω0\in \Omega is an exposed point for {ϕ=0}={ϕ≤0}\left\{\phi =0\right\}=\left\{\phi \le 0\right\}.Now, let ϕε(x)≔ϕ(x)−ε(x1+1){\phi }_{\varepsilon }\left(x):= \phi \left(x)-\varepsilon \left({x}_{1}+1)and define S0≔{ϕ=0}∩{x1≥−1}andSε≔{ϕε≤0}.{\pmb{\mathscr{S}}}_{0}:= \left\{\phi =0\right\}\cap \left\{{x}_{1}\ge -1\right\}\hspace{1.0em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1.0em}{\pmb{\mathscr{S}}}_{\varepsilon }:= \left\{{\phi }_{\varepsilon }\le 0\right\}.Also, let γε{\gamma }_{\varepsilon }be defined by γε≔(Id−εe1)#γ.{\gamma }_{\varepsilon }:= {\left({\rm{Id}}-\varepsilon {{\bf{e}}}_{1})}_{\#}\gamma .Notice that, by construction, S0{\pmb{\mathscr{S}}}_{0}is compact, S0⊂{x1≤0}{\pmb{\mathscr{S}}}_{0}\subset \left\{{x}_{1}\le 0\right\}, 0,−e1∈S00,-{{\bf{e}}}_{1}\in {\pmb{\mathscr{S}}}_{0}, and Sε→S0{\pmb{\mathscr{S}}}_{\varepsilon }\to {\pmb{\mathscr{S}}}_{0}in the Hausdorff sense as ε→0\varepsilon \to 0; in particular, there exists D>0\pmb{\mathscr{D}}\gt 0such that Sε⊂BDfor allε≪1.{\pmb{\mathscr{S}}}_{\varepsilon }\subset {\pmb{\mathscr{B}}}_{\pmb{\mathscr{D}}}\hspace{1.0em}\hspace{0.1em}\text{for all}\hspace{0.1em}\hspace{0.33em}\varepsilon \ll 1.Also, if aε>0{a}_{\varepsilon }\gt 0is such that Πε≔{x1=aε}{\Pi }_{\varepsilon }:= \left\{{x}_{1}={a}_{\varepsilon }\right\}is a supporting plane to Sε{\pmb{\mathscr{S}}}_{\varepsilon }, we see that aε→0asε→0{a}_{\varepsilon }\to 0\hspace{1.0em}\hspace{0.1em}\text{as}\hspace{0.1em}\hspace{0.33em}\varepsilon \to 0and ε=∣ϕε(0)∣≤maxSε∣ϕε∣≤(1+aε)ε.\varepsilon =| {\phi }_{\varepsilon }\left(0)| \le \mathop{\max }\limits_{{\pmb{\mathscr{S}}}_{\varepsilon }}| {\phi }_{\varepsilon }| \le \left(1+{a}_{\varepsilon })\varepsilon .Let Aε{\pmb{\mathscr{A}}}_{\varepsilon }be the John transformation (affine map) that normalizes Sε{\pmb{\mathscr{S}}}_{\varepsilon }[6]: Aεx≔Lε(x−xε),{\pmb{\mathscr{A}}}_{\varepsilon }x:= {\pmb{\mathscr{L}}}_{\varepsilon }\left(x-{x}_{\varepsilon }),where xε{x}_{\varepsilon }is the center of mass of Sε{\pmb{\mathscr{S}}}_{\varepsilon }and Lε:Rn→Rn{\pmb{\mathscr{L}}}_{\varepsilon }:{{\mathbb{R}}}^{n}\to {{\mathbb{R}}}^{n}is a symmetric and positive definite linear transformation. Set ϕ˜ε(x)≔ϕε(Aε−1x)εandS˜ε≔Aε(Sε).{\tilde{\phi }}_{\varepsilon }\left(x):= \frac{{\phi }_{\varepsilon }\left({\pmb{\mathscr{A}}}_{\varepsilon }^{-1}x)}{\varepsilon }\hspace{1.0em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1.0em}{\tilde{\pmb{\mathscr{S}}}}_{\varepsilon }:= {\pmb{\mathscr{A}}}_{\varepsilon }\left({\pmb{\mathscr{S}}}_{\varepsilon }).Then B1⊂S˜ε⊂Bn3/2{\pmb{\mathscr{B}}}_{1}\subset {\tilde{\pmb{\mathscr{S}}}}_{\varepsilon }\subset {\pmb{\mathscr{B}}}_{{n}^{3\text{/}2}}and 1=∣ϕ˜ε(0˜ε)∣≤maxS˜ε∣ϕ˜ε∣≤1+aεwith0˜ε≔Aε(0).1=| {\tilde{\phi }}_{\varepsilon }\left({\tilde{0}}_{\varepsilon })| \le \mathop{\max }\limits_{{\tilde{\pmb{\mathscr{S}}}}_{\varepsilon }}| {\tilde{\phi }}_{\varepsilon }| \le 1+{a}_{\varepsilon }\hspace{1.0em}\hspace{0.1em}\text{with}\hspace{0.1em}\hspace{0.33em}{\tilde{0}}_{\varepsilon }:= {\pmb{\mathscr{A}}}_{\varepsilon }\left(0).Recall that affine transformations preserve the ratio of the distances between parallel planes; therefore, letting Π−1≔{x1=−1}{\Pi }_{-1}:= \left\{{x}_{1}=-1\right\}, Π0≔{x1=0}{\Pi }_{0}:= \left\{{x}_{1}=0\right\}, and Π˜i≔Aε(Πi){\tilde{\Pi }}_{i}:= {\pmb{\mathscr{A}}}_{\varepsilon }\left({\Pi }_{i})for i=−1,0,εi=-1,0,\varepsilon , we have that dist(Π˜0,Π˜ε)dist(Π˜−1,Π˜ε)=dist(Π0,Πε)dist(Π−1,Πε)=aε1+aε.\frac{{\rm{dist}}\left({\tilde{\Pi }}_{0},{\tilde{\Pi }}_{\varepsilon })}{{\rm{dist}}\left({\tilde{\Pi }}_{-1},{\tilde{\Pi }}_{\varepsilon })}=\frac{{\rm{dist}}\left({\Pi }_{0},{\Pi }_{\varepsilon })}{{\rm{dist}}\left({\Pi }_{-1},{\Pi }_{\varepsilon })}=\frac{{a}_{\varepsilon }}{1+{a}_{\varepsilon }}.In turn, dist(0˜ε,∂S˜ε)≤dist(Π˜0,Π˜ε)≤dist(Π˜−1,Π˜ε)aε1+aε≤diam(S˜ε)aε≤2n3/2aε,{\rm{dist}}\left({\tilde{0}}_{\varepsilon },\partial {\tilde{\pmb{\mathscr{S}}}}_{\varepsilon })\le {\rm{dist}}\left({\tilde{\Pi }}_{0},{\tilde{\Pi }}_{\varepsilon })\le {\rm{dist}}\left({\tilde{\Pi }}_{-1},{\tilde{\Pi }}_{\varepsilon })\frac{{a}_{\varepsilon }}{1+{a}_{\varepsilon }}\le {\rm{diam}}\left({\tilde{\pmb{\mathscr{S}}}}_{\varepsilon }){a}_{\varepsilon }\le 2{n}^{3\text{/}2}{a}_{\varepsilon },and considering the cone generated by ∂S˜ε\partial {\tilde{\pmb{\mathscr{S}}}}_{\varepsilon }over (0˜ε,ϕ˜ε(0˜ε))\left({\tilde{0}}_{\varepsilon },{\tilde{\phi }}_{\varepsilon }\left({\tilde{0}}_{\varepsilon })), we find that Kε≔convBrn∪rnaεe1⊂∂ϕ˜ε(S˜ε)withrn≔12n3/2.{\pmb{\mathscr{K}}}_{\varepsilon }:= {\rm{conv}}\left({\pmb{\mathscr{B}}}_{{r}_{n}}\cup \left\{\phantom{\rule[-1.25em]{}{0ex}},\frac{{r}_{n}}{{a}_{\varepsilon }}{{\bf{e}}}_{1}\right\}\right)\subset \partial {\tilde{\phi }}_{\varepsilon }\left({\tilde{\pmb{\mathscr{S}}}}_{\varepsilon })\hspace{1.0em}\hspace{0.1em}\text{with}\hspace{0.1em}\hspace{0.33em}{r}_{n}:= \frac{1}{2{n}^{3\text{/}2}}.(For more details on this inclusion, see, e.g., [7, Theorem 2.8].) So if we let ρ˜ε≔(Aε)#ρandγ˜ε≔(ε−1Lε−1)#γε,{\tilde{\rho }}_{\varepsilon }:= {\left({\pmb{\mathscr{A}}}_{\varepsilon })}_{\#}\rho \hspace{1.0em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1.0em}{\tilde{\gamma }}_{\varepsilon }:= {\left({\varepsilon }^{-1}{\pmb{\mathscr{L}}}_{\varepsilon }^{-1})}_{\#}{\gamma }_{\varepsilon },then (∇ϕ˜ε)#ρ˜ε=γ˜ε{\left(\nabla {\tilde{\phi }}_{\varepsilon })}_{\#}{\tilde{\rho }}_{\varepsilon }={\tilde{\gamma }}_{\varepsilon }, and, by the mass balance formula, (2.3)γ˜ε(Kε)≤γ˜ε(∂ϕ˜ε(S˜ε))=ρ˜ε(S˜ε)≤ρ˜ε(Bn3/2).{\tilde{\gamma }}_{\varepsilon }\left({\pmb{\mathscr{K}}}_{\varepsilon })\le {\tilde{\gamma }}_{\varepsilon }(\partial {\tilde{\phi }}_{\varepsilon }\left({\tilde{\pmb{\mathscr{S}}}}_{\varepsilon }))={\tilde{\rho }}_{\varepsilon }\left({\tilde{\pmb{\mathscr{S}}}}_{\varepsilon })\le {\tilde{\rho }}_{\varepsilon }\left({\pmb{\mathscr{B}}}_{{n}^{3\text{/}2}}).On the other hand, since Sε⊂BD{\pmb{\mathscr{S}}}_{\varepsilon }\subset {\pmb{\mathscr{B}}}_{\pmb{\mathscr{D}}}and S˜ε⊃B1{\tilde{\pmb{\mathscr{S}}}}_{\varepsilon }\supset {\pmb{\mathscr{B}}}_{1}for ε≪1\varepsilon \ll 1, we see that ∣Aε(w)−Aε(z)∣≥1D∣w−z∣for allw,z∈Rn.| {\pmb{\mathscr{A}}}_{\varepsilon }\left(w)-{\pmb{\mathscr{A}}}_{\varepsilon }\left(z)| \ge \frac{1}{\pmb{\mathscr{D}}}| w-z| \hspace{1.0em}\hspace{0.1em}\text{for all}\hspace{0.1em}\hspace{0.33em}w,z\in {{\mathbb{R}}}^{n}.In turn, for ε≪1\varepsilon \ll 1, Ω˜ε≔Aε(Ω)⊃Aε(Bd(−e1))⊃BdD(Aε(−e1)).{\tilde{\Omega }}_{\varepsilon }:= {\pmb{\mathscr{A}}}_{\varepsilon }\left(\Omega )\supset {\pmb{\mathscr{A}}}_{\varepsilon }\left({\pmb{\mathscr{B}}}_{d}\left(-{{\bf{e}}}_{1}))\supset {\pmb{\mathscr{B}}}_{\tfrac{d}{\pmb{\mathscr{D}}}}\left({\pmb{\mathscr{A}}}_{\varepsilon }\left(-{{\bf{e}}}_{1})).Therefore, if we define S˜ε,d≔dist(⋅,∂S˜ε)≥d2D,{\tilde{\pmb{\mathscr{S}}}}_{\varepsilon ,d}:= \left\{\phantom{\rule[-1.25em]{}{0ex}},{\rm{dist}}\left(\cdot ,\partial {\tilde{\pmb{\mathscr{S}}}}_{\varepsilon })\ge \frac{d}{2\pmb{\mathscr{D}}}\right\},then there exists a dimensional constant Cn>0{\pmb{\mathscr{C}}}_{n}\gt 0and a point z˜d{\tilde{z}}_{d}such that Aε(Bd(−e1))∩S˜ε,d⊃BdCnD(z˜d).{\pmb{\mathscr{A}}}_{\varepsilon }\left({\pmb{\mathscr{B}}}_{d}\left(-{{\bf{e}}}_{1}))\cap {\tilde{\pmb{\mathscr{S}}}}_{\varepsilon ,d}\supset {\pmb{\mathscr{B}}}_{\tfrac{d}{{\pmb{\mathscr{C}}}_{n}\pmb{\mathscr{D}}}}\left({\tilde{z}}_{d}).Also, by, for example, [7, Corollary A.23], ∂ϕ˜ε(S˜ε,d)⊂B6Dd.\partial {\tilde{\phi }}_{\varepsilon }\left({\tilde{\pmb{\mathscr{S}}}}_{\varepsilon ,d})\subset {\pmb{\mathscr{B}}}_{\tfrac{6\pmb{\mathscr{D}}}{d}}.Thus, for all ε≪1\varepsilon \ll 1, (2.4)ρ˜ε(Bn3/2)≤μ˜ε(B2n3/2(z˜d))≤Cμkμ˜εBdCnD(z˜d)=Cμkρ˜εBdCnD(z˜d),{\tilde{\rho }}_{\varepsilon }\left({\pmb{\mathscr{B}}}_{{n}^{3\text{/}2}})\le {\tilde{\mu }}_{\varepsilon }\left({\pmb{\mathscr{B}}}_{2{n}^{3\text{/}2}}\left({\tilde{z}}_{d}))\le {\pmb{\mathscr{C}}}_{\mu }^{k}{\tilde{\mu }}_{\varepsilon }\left({\pmb{\mathscr{B}}}_{\tfrac{d}{{\pmb{\mathscr{C}}}_{n}\pmb{\mathscr{D}}}}\left({\tilde{z}}_{d})\right)={\pmb{\mathscr{C}}}_{\mu }^{k}\hspace{0.16em}{\tilde{\rho }}_{\varepsilon }\left({\pmb{\mathscr{B}}}_{\tfrac{d}{{\pmb{\mathscr{C}}}_{n}\pmb{\mathscr{D}}}}\left({\tilde{z}}_{d})\right),where μ˜ε≔(Aε)#μ{\tilde{\mu }}_{\varepsilon }:= {\left({\pmb{\mathscr{A}}}_{\varepsilon })}_{\#}\mu , the number k∈Nk\in {\mathbb{N}}is such that 2n3/2≤2kdCnD2{n}^{3\text{/}2}\le {2}^{k}\frac{d}{{\pmb{\mathscr{C}}}_{n}\pmb{\mathscr{D}}}, and Cμ{\pmb{\mathscr{C}}}_{\mu }is the doubling constant for μ\mu in B4Dn3/2{\pmb{\mathscr{B}}}_{4\pmb{\mathscr{D}}{n}^{3\text{/}2}}. (The last equality holds since μ˜ε{\tilde{\mu }}_{\varepsilon }and ρ˜ε{\tilde{\rho }}_{\varepsilon }agree on Ω˜ε{\tilde{\Omega }}_{\varepsilon }.) Moreover, by using the mass balance formula again, we deduce that (2.5)ρ˜εBdCnD(z˜d)≤ρ˜ε(Aε(Bd(−e1))∩S˜ε,d)=γ˜ε(∂ϕ˜ε(Aε(Bd(−e1))∩S˜ε,d))≤γ˜εB6Dd{\tilde{\rho }}_{\varepsilon }\left({\pmb{\mathscr{B}}}_{\tfrac{d}{{\pmb{\mathscr{C}}}_{n}\pmb{\mathscr{D}}}}\left({\tilde{z}}_{d})\right)\le {\tilde{\rho }}_{\varepsilon }({\pmb{\mathscr{A}}}_{\varepsilon }\left({\pmb{\mathscr{B}}}_{d}\left(-{{\bf{e}}}_{1}))\cap {\tilde{\pmb{\mathscr{S}}}}_{\varepsilon ,d})={\tilde{\gamma }}_{\varepsilon }(\partial {\tilde{\phi }}_{\varepsilon }\left({\pmb{\mathscr{A}}}_{\varepsilon }\left({\pmb{\mathscr{B}}}_{d}\left(-{{\bf{e}}}_{1}))\cap {\tilde{\pmb{\mathscr{S}}}}_{\varepsilon ,d}))\le {\tilde{\gamma }}_{\varepsilon }\left({\pmb{\mathscr{B}}}_{\tfrac{6\pmb{\mathscr{D}}}{d}}\right)for all ε≪1\varepsilon \ll 1. Consequently, combining the three chains of inequalities (2.3), (2.4), and (2.5), we have that (2.6)γ˜ε(Kε)≤Cμkγ˜εB6Dd.{\tilde{\gamma }}_{\varepsilon }\left({\pmb{\mathscr{K}}}_{\varepsilon })\le {\pmb{\mathscr{C}}}_{\mu }^{k}{\tilde{\gamma }}_{\varepsilon }\left({\pmb{\mathscr{B}}}_{\tfrac{6\pmb{\mathscr{D}}}{d}}\right).Now, let tme1∈Kε{t}_{m}{{\bf{e}}}_{1}\in {\pmb{\mathscr{K}}}_{\varepsilon }for m=1,…,Mm=1,\ldots ,\pmb{\mathscr{M}}be a sequence of points chosenA possible way to construct such a sequence is to choose tm=5m{t}_{m}={5}^{m}. To ensure that tme1∈Kε{t}_{m}{{\bf{e}}}_{1}\in {\pmb{\mathscr{K}}}_{\varepsilon }for any m=1,…,Mm=1,\ldots ,\pmb{\mathscr{M}}, one needs M≤logrn−logaεlog5\pmb{\mathscr{M}}\le \frac{\log {r}_{n}-\log {a}_{\varepsilon }}{\log 5}.so that 12Km⊂Km⧹Km−1withKm≔conv(Brn∪{tme1})andK0≔Brn.\frac{1}{2}{\pmb{\mathscr{K}}}_{m}\subset {\pmb{\mathscr{K}}}_{m}\setminus {\pmb{\mathscr{K}}}_{m-1}\hspace{1.0em}\hspace{0.1em}\text{with}\hspace{0.1em}\hspace{0.33em}{\pmb{\mathscr{K}}}_{m}:= {\rm{conv}}\left({\pmb{\mathscr{B}}}_{{r}_{n}}\cup \left\{{t}_{m}{{\bf{e}}}_{1}\right\})\hspace{1.0em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1.0em}{\pmb{\mathscr{K}}}_{0}:= {\pmb{\mathscr{B}}}_{{r}_{n}}.By construction, 12Kmm=1M{\left\{\frac{1}{2}{\pmb{\mathscr{K}}}_{m}\right\}}_{m=1}^{\pmb{\mathscr{M}}}is a disjoint family, and (2.7)M=M(aε)→∞asaε→0.\pmb{\mathscr{M}}=\pmb{\mathscr{M}}\left({a}_{\varepsilon })\to \infty \hspace{1.0em}\hspace{0.1em}\text{as}\hspace{0.1em}\hspace{0.33em}{a}_{\varepsilon }\to 0.Hence, since εLε(Kε)⊂ϒ\varepsilon {\pmb{\mathscr{L}}}_{\varepsilon }\left({\pmb{\mathscr{K}}}_{\varepsilon })\subset \Upsilon , we find that Mγ˜ε(Brn)≤∑m=1Mγ˜ε(Km)≤Cγ∑m=1Mγ˜ε12Km≤Cγγ˜ε(Kε),\pmb{\mathscr{M}}{\tilde{\gamma }}_{\varepsilon }\left({\pmb{\mathscr{B}}}_{{r}_{n}})\le \mathop{\sum }\limits_{m=1}^{\pmb{\mathscr{M}}}{\tilde{\gamma }}_{\varepsilon }\left({\pmb{\mathscr{K}}}_{m})\le {\pmb{\mathscr{C}}}_{\gamma }\mathop{\sum }\limits_{m=1}^{\pmb{\mathscr{M}}}{\tilde{\gamma }}_{\varepsilon }\left(\frac{1}{2}{\pmb{\mathscr{K}}}_{m}\right)\le {\pmb{\mathscr{C}}}_{\gamma }{\tilde{\gamma }}_{\varepsilon }\left({\pmb{\mathscr{K}}}_{\varepsilon }),with Cγ{\pmb{\mathscr{C}}}_{\gamma }denoting the doubling constant for γ\gamma in B2R{\pmb{\mathscr{B}}}_{2\pmb{\mathscr{R}}}, which is the same as the doubling constant for ν\nu in B2R{\pmb{\mathscr{B}}}_{2\pmb{\mathscr{R}}}; since ϒ\Upsilon is convex, γ\gamma inherits its doubling property from ν\nu . All in all, considering the aforementioned chain of inequalities and (2.6) and denoting by j∈Nj\in {\mathbb{N}}the smallest number such that 6Dd≤2jrn\frac{6\pmb{\mathscr{D}}}{d}\le {2}^{j}{r}_{n}, we see that 0<Mγ˜ε(Brn)≤CμkCγγ˜εB6Dd≤CμkCγj+1γ˜ε(Brn),0\lt \pmb{\mathscr{M}}{\tilde{\gamma }}_{\varepsilon }\left({\pmb{\mathscr{B}}}_{{r}_{n}})\le {\pmb{\mathscr{C}}}_{\mu }^{k}{\pmb{\mathscr{C}}}_{\gamma }{\tilde{\gamma }}_{\varepsilon }\left({\pmb{\mathscr{B}}}_{\tfrac{6\pmb{\mathscr{D}}}{d}}\right)\le {\pmb{\mathscr{C}}}_{\mu }^{k}{\pmb{\mathscr{C}}}_{\gamma }^{j+1}{\tilde{\gamma }}_{\varepsilon }\left({\pmb{\mathscr{B}}}_{{r}_{n}}),or, equivalently, M≤CμkCγj+1.\pmb{\mathscr{M}}\le {\pmb{\mathscr{C}}}_{\mu }^{k}{\pmb{\mathscr{C}}}_{\gamma }^{j+1}.But this is impossible for small ε\varepsilon , concluding the proof.Case 2b: 0∈∂(dom(u))∩X¯0\in \partial \left({\rm{dom}}\left(u))\cap \overline{\pmb{\mathscr{X}}}. In this subcase, let uε(x)≔u(x)−ε(x1+1){u}_{\varepsilon }\left(x):= u\left(x)-\varepsilon \left({x}_{1}+1)and define S0≔Σ∩{x1≥−1}andSε≔{uε≤0}.{\pmb{\mathscr{S}}}_{0}:= \Sigma \cap \left\{{x}_{1}\ge -1\right\}\hspace{1.0em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1.0em}{\pmb{\mathscr{S}}}_{\varepsilon }:= \left\{{u}_{\varepsilon }\le 0\right\}.Like before, for all ε≪1\varepsilon \ll 1, Sε⊂BD{\pmb{\mathscr{S}}}_{\varepsilon }\subset {\pmb{\mathscr{B}}}_{\pmb{\mathscr{D}}}for some D>0\pmb{\mathscr{D}}\gt 0. Here, however, as 0∈∂(dom(u))0\in \partial \left({\rm{dom}}\left(u)), we have that ∂u˜ε(S˜ε)⊃conv(Brn∪R+e1)withrn≔12n3/2.\partial {\tilde{u}}_{\varepsilon }\left({\tilde{\pmb{\mathscr{S}}}}_{\varepsilon })\supset {\rm{conv}}\left({\pmb{\mathscr{B}}}_{{r}_{n}}\cup {{\mathbb{R}}}^{+}{{\bf{e}}}_{1})\hspace{1.0em}\hspace{0.1em}\text{with}\hspace{0.1em}\hspace{0.33em}{r}_{n}:= \frac{1}{2{n}^{3\text{/}2}}.The function u˜ε{\tilde{u}}_{\varepsilon }is defined in an analogous fashion to how ϕ˜ε{\tilde{\phi }}_{\varepsilon }was defined in Case 2a (but replacing ϕ\phi by uu) and, again, S˜ε≔Aε(Sε){\tilde{\pmb{\mathscr{S}}}}_{\varepsilon }:= {\pmb{\mathscr{A}}}_{\varepsilon }\left({\pmb{\mathscr{S}}}_{\varepsilon })with Aε{\pmb{\mathscr{A}}}_{\varepsilon }denoting the John map associated to Sε{\pmb{\mathscr{S}}}_{\varepsilon }whose linear part is Lε{\pmb{\mathscr{L}}}_{\varepsilon }. In turn, arguing as we did in Case 2a, where again k∈Nk\in {\mathbb{N}}is such that 2n3/2≤2kdCnD2{n}^{3\text{/}2}\le {2}^{k}\frac{d}{{\pmb{\mathscr{C}}}_{n}\pmb{\mathscr{D}}}and Cμ{\pmb{\mathscr{C}}}_{\mu }is the doubling constant for μ\mu in B4Dn3/2{\pmb{\mathscr{B}}}_{4\pmb{\mathscr{D}}{n}^{3\text{/}2}}, but in the original variables, we deduce that νε(εLε(Kε))≤CμkνεεLεB6Dd{\nu }_{\varepsilon }(\varepsilon {\pmb{\mathscr{L}}}_{\varepsilon }\left({\pmb{\mathscr{K}}}_{\varepsilon }))\le {\pmb{\mathscr{C}}}_{\mu }^{k}{\nu }_{\varepsilon }\left(\varepsilon {\pmb{\mathscr{L}}}_{\varepsilon }\left({\pmb{\mathscr{B}}}_{\tfrac{6\pmb{\mathscr{D}}}{d}}\right)\right)for all ε≪1\varepsilon \ll 1(cf. (2.6)). Here, instead, νε≔(Id−εe1)#ν{\nu }_{\varepsilon }:= {\left({\rm{Id}}-\varepsilon {{\bf{e}}}_{1})}_{\#}\nu and Kε≔convBrn∪rnεe1.{\pmb{\mathscr{K}}}_{\varepsilon }:= {\rm{conv}}\left({\pmb{\mathscr{B}}}_{{r}_{n}}\cup \left\{\frac{{r}_{n}}{\varepsilon }{{\bf{e}}}_{1}\right\}\right).Now notice that ∣Lε(e1)∣=∣Aε(0)−Aε(−e1)∣≤2n3/2.| {\pmb{\mathscr{L}}}_{\varepsilon }\left({{\bf{e}}}_{1})| =| {\pmb{\mathscr{A}}}_{\varepsilon }\left(0)-{\pmb{\mathscr{A}}}_{\varepsilon }\left(-{{\bf{e}}}_{1})| \le 2{n}^{3\text{/}2}.Moreover, we claim there exists an N≫2n3/2>0\pmb{\mathscr{N}}\gg 2{n}^{3\text{/}2}\gt 0such that ‖εLε‖≤Nfor allε≪1.\Vert \varepsilon {\pmb{\mathscr{L}}}_{\varepsilon }\Vert \le \pmb{\mathscr{N}}\hspace{1.0em}\hspace{0.1em}\text{for all}\hspace{0.1em}\hspace{0.33em}\varepsilon \ll 1.Indeed, if not, then we can find a sequence of points zε∈Sε{z}_{\varepsilon }\in {\pmb{\mathscr{S}}}_{\varepsilon }and slopes pε∈∂uε(zε)∩span(S0)⊥{p}_{\varepsilon }\in \partial {u}_{\varepsilon }\left({z}_{\varepsilon })\cap {\rm{span}}{\left({\pmb{\mathscr{S}}}_{0})}^{\perp }such that ∣pε∣→∞| {p}_{\varepsilon }| \to \infty . In particular, in the limit, we find a point z0∈S0{z}_{0}\in {\pmb{\mathscr{S}}}_{0}such that ∂u(z0)∩span(S0)⊥\partial u\left({z}_{0})\cap {\rm{span}}{\left({\pmb{\mathscr{S}}}_{0})}^{\perp }contains a sequence of slopes {pj}j∈N{\left\{{p}_{j}\right\}}_{j\in {\mathbb{N}}}with ∣pj∣=j| {p}_{j}| =j. But as pj∈span(S0)⊥{p}_{j}\in {\rm{span}}{\left({\pmb{\mathscr{S}}}_{0})}^{\perp }, we see that pj⋅(x−z0)=pj⋅x=pj⋅(x−z){p}_{j}\cdot \left(x-{z}_{0})={p}_{j}\cdot x={p}_{j}\cdot \left(x-z)for any z∈S0z\in {\pmb{\mathscr{S}}}_{0}. Hence, pj∈∂u(z){p}_{j}\in \partial u\left(z)for all z∈S0z\in {\pmb{\mathscr{S}}}_{0}and j∈Nj\in {\mathbb{N}}. However, this is impossible; S0∩int(dom(u)){\pmb{\mathscr{S}}}_{0}\cap {\rm{int}}\left({\rm{dom}}\left(u))is nonempty, and on this set, uuis locally Lipschitz, proving the claim.Therefore, εLε(Kε)⊂BNandεLεB6Dd⊂B6DNd.\varepsilon {\pmb{\mathscr{L}}}_{\varepsilon }\left({\pmb{\mathscr{K}}}_{\varepsilon })\subset {\pmb{\mathscr{B}}}_{\pmb{\mathscr{N}}}\hspace{1.0em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1.0em}\varepsilon {\pmb{\mathscr{L}}}_{\varepsilon }\left({\pmb{\mathscr{B}}}_{\tfrac{6\pmb{\mathscr{D}}}{d}}\right)\subset {\pmb{\mathscr{B}}}_{\tfrac{6\pmb{\mathscr{D}}\pmb{\mathscr{N}}}{d}}.And so, arguing exactly like we did in Case 2a, we find that M≤CμkCνj+1,\pmb{\mathscr{M}}\le {\pmb{\mathscr{C}}}_{\mu }^{k}{\pmb{\mathscr{C}}}_{\nu }^{j+1},where Cν{\pmb{\mathscr{C}}}_{\nu }is the doubling constant for ν\nu in B6DN/d{\pmb{\mathscr{B}}}_{6\pmb{\mathscr{D}}\pmb{\mathscr{N}}\text{/}d}and M=M(ε)→∞\pmb{\mathscr{M}}=\pmb{\mathscr{M}}\left(\varepsilon )\to \infty as ε→0\varepsilon \to 0is the analogous count for this case’s Kε{\pmb{\mathscr{K}}}_{\varepsilon }(cf. (2.7)). But, again, this is impossible.Case 3. Σ\Sigma has an exposed point xˆ\hat{x}in Rn⧹X¯{{\mathbb{R}}}^{n}\setminus \overline{\pmb{\mathscr{X}}}. In this case, up to a translation, a dilation, a rotation, and subtracting ℓ\ell from uu, we can assume that xˆ=0,Σ⊂{x1≤0},ℓ≡0,andS0≔Σ∩{x1≥−1}⋐Rn⧹X¯.\hat{x}=0,\hspace{1.0em}\Sigma \subset \left\{{x}_{1}\le 0\right\},\hspace{1.0em}\ell \equiv 0,\hspace{1.0em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1.0em}{\pmb{\mathscr{S}}}_{0}:= \Sigma \cap \left\{{x}_{1}\ge -1\right\}\hspace{0.33em}\Subset \hspace{0.33em}{{\mathbb{R}}}^{n}\setminus \overline{\pmb{\mathscr{X}}}.Like before, let uε(x)≔u(x)−ε(x1+1){u}_{\varepsilon }\left(x):= u\left(x)-\varepsilon \left({x}_{1}+1)and define Sε≔{uε≤0}andνε≔(Id−εe1)#ν.{\pmb{\mathscr{S}}}_{\varepsilon }:= \left\{{u}_{\varepsilon }\le 0\right\}\hspace{1.0em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1.0em}{\nu }_{\varepsilon }:= {\left({\rm{Id}}-\varepsilon {{\bf{e}}}_{1})}_{\#}\nu .Again, Sε→S0{\pmb{\mathscr{S}}}_{\varepsilon }\to {\pmb{\mathscr{S}}}_{0}as ε→0\varepsilon \to 0, so diam(Sε)≤2diam(S0)andSε⋐Rn⧹X¯{\rm{diam}}\left({\pmb{\mathscr{S}}}_{\varepsilon })\le 2{\rm{diam}}\left({\pmb{\mathscr{S}}}_{0})\hspace{1.0em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1.0em}{\pmb{\mathscr{S}}}_{\varepsilon }\hspace{0.33em}\Subset \hspace{0.33em}{{\mathbb{R}}}^{n}\setminus \overline{\pmb{\mathscr{X}}}for all ε≪1\varepsilon \ll 1. For these small positive ε\varepsilon , then, 0=μ(Sε)=νε(∂uε(Sε)),0=\mu \left({\pmb{\mathscr{S}}}_{\varepsilon })={\nu }_{\varepsilon }\left(\partial {u}_{\varepsilon }\left({\pmb{\mathscr{S}}}_{\varepsilon })),where the second equality follows from the mass balance formula. (Recall that μ\mu vanishes on Rn⧹X¯{{\mathbb{R}}}^{n}\setminus \overline{\pmb{\mathscr{X}}}.) Moreover, as Y\pmb{\mathscr{Y}}is convex, ∂uε(Sε)⊂spt(νε)=Y¯−εe1\partial {u}_{\varepsilon }\left({\pmb{\mathscr{S}}}_{\varepsilon })\subset {\rm{spt}}\left({\nu }_{\varepsilon })=\overline{\pmb{\mathscr{Y}}}-\varepsilon {{\bf{e}}}_{1}(cf. (2.1)). Thus, any open subset of ∂uε(Sε)\partial {u}_{\varepsilon }\left({\pmb{\mathscr{S}}}_{\varepsilon })must be in the interior of the support of νε{\nu }_{\varepsilon }. In turn, considering the cone generated by ∂Sε\partial {\pmb{\mathscr{S}}}_{\varepsilon }over (0,uε(0))\left(0,{u}_{\varepsilon }\left(0)), for 0<ε≪10\lt \varepsilon \ll 1, we find that 0=νε(∂uε(Sε))≥νε(Brε)>0withrε≔∣uε(0)∣2diam(S0).0={\nu }_{\varepsilon }\left(\partial {u}_{\varepsilon }\left({\pmb{\mathscr{S}}}_{\varepsilon }))\ge {\nu }_{\varepsilon }\left({\pmb{\mathscr{B}}}_{{r}_{\varepsilon }})\gt 0\hspace{1.0em}\hspace{0.1em}\text{with}\hspace{0.1em}\hspace{0.33em}{r}_{\varepsilon }:= \frac{| {u}_{\varepsilon }\left(0)| }{2{\rm{diam}}\left({\pmb{\mathscr{S}}}_{0})}.(Again, for more details on this inclusion, see, e.g., [7, Theorem 2.8].) This is a contradiction and concludes the proof.□3Proof of Theorem 1.6Again, we replace uuby its lower-semicontinuous extension outside of X\pmb{\mathscr{X}}, exactly as we did at the beginning of Section 2. We split the proof into three parts.Part 1. uuis continuously differentiable inside X\pmb{\mathscr{X}}.We follow the argument used to prove [2, Corollary 1]. Assume for the sake of a contradiction that the result is false. Up to a translation, let 0∈X0\in \pmb{\mathscr{X}}be a point at which uuhas two distinct supporting planes. After a rotation, dilation, and subtracting off an affine function from uu, we may assume that u(x)≥max{x1,0},u(0)=0,andu(−se1)s→0ass→0.u\left(x)\ge \max \left\{{x}_{1},0\right\},\hspace{1.0em}u\left(0)=0,\hspace{1.0em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1.0em}\frac{u\left(-s{{\bf{e}}}_{1})}{s}\to 0\hspace{1em}\hspace{0.1em}\text{as}\hspace{0.1em}\hspace{0.33em}s\to 0.Now consider the function uσ{u}_{\sigma }defined by uσ(x)≔u(x)−τ(x1+2σ)withτ≔u(−σe1)σ.{u}_{\sigma }\left(x):= u\left(x)-\tau \left({x}_{1}+2\sigma )\hspace{1em}\hspace{0.1em}\text{with}\hspace{0.1em}\hspace{0.33em}\tau := \frac{u\left(-\sigma {{\bf{e}}}_{1})}{\sigma }.Note that τ→0\tau \to 0as σ→0\sigma \to 0. If Sσ≔{uσ≤0},{\pmb{\mathscr{S}}}_{\sigma }:= \left\{{u}_{\sigma }\le 0\right\},then, by the strict convexity of uuprovided by Theorem 1.2, we see that Sσ⊂BD⋐X∩int(dom(u)){\pmb{\mathscr{S}}}_{\sigma }\subset {\pmb{\mathscr{B}}}_{\pmb{\mathscr{D}}}\hspace{0.33em}\Subset \hspace{0.33em}\pmb{\mathscr{X}}\cap {\rm{int}}\left({\rm{dom}}\left(u))for some D>0\pmb{\mathscr{D}}\gt 0and for all σ≪1\sigma \ll 1; also, for these small positive σ\sigma , ∂uσ(Sσ)⊂BR∩Y\partial {u}_{\sigma }\left({\pmb{\mathscr{S}}}_{\sigma })\subset {\pmb{\mathscr{B}}}_{\pmb{\mathscr{R}}}\cap \pmb{\mathscr{Y}}for some R>0\pmb{\mathscr{R}}\gt 0. Moreover, if Π−a≔{x1=−a}{\Pi }_{-a}:= \left\{{x}_{1}=-a\right\}and Πb≔{x1=b}{\Pi }_{b}:= \left\{{x}_{1}=b\right\}denote the two parallel planes that tangentially sandwich Sσ{\pmb{\mathscr{S}}}_{\sigma }, we see that a>σandb<2τσ1−τ,a\gt \sigma \hspace{1.0em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1.0em}b\lt \frac{2\tau \sigma }{1-\tau },provided σ>0\sigma \gt 0is small enough to guarantee that τ<1\tau \lt 1. Furthermore, maxSσ∣uσ∣=∣uσ(0)∣=2τσ,\mathop{\max }\limits_{{\pmb{\mathscr{S}}}_{\sigma }}| {u}_{\sigma }| =| {u}_{\sigma }\left(0)| =2\tau \sigma ,and dist(Πb,Π0)dist(Π−a,Πb)=ba+b≤ba≤2τ1−τ→0asσ→0,\frac{{\rm{dist}}\left({\Pi }_{b},{\Pi }_{0})}{{\rm{dist}}\left({\Pi }_{-a},{\Pi }_{b})}=\frac{b}{a+b}\le \frac{b}{a}\le \frac{2\tau }{1-\tau }\to 0\hspace{1.0em}\hspace{0.1em}\text{as}\hspace{0.1em}\hspace{0.33em}\sigma \to 0,where Π0≔{x1=0}{\Pi }_{0}:= \left\{{x}_{1}=0\right\}.Now, set u˜σ(x)≔uσ(Aσ−1x)2τσandS˜σ≔Aσ(Sσ),{\tilde{u}}_{\sigma }\left(x):= \frac{{u}_{\sigma }\left({\pmb{\mathscr{A}}}_{\sigma }^{-1}x)}{2\tau \sigma }\hspace{1.0em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1.0em}{\tilde{\pmb{\mathscr{S}}}}_{\sigma }:= {\pmb{\mathscr{A}}}_{\sigma }\left({\pmb{\mathscr{S}}}_{\sigma }),where Aσ{\pmb{\mathscr{A}}}_{\sigma }is the John map that normalizes Sσ{\pmb{\mathscr{S}}}_{\sigma }. Arguing as we did in the proof of Theorem 1.2, we find the same contradiction as we did in Case 2 when τ\tau sufficiently small; the only difference is that we consider a slightly different chain of inequalities: ν˜σ(Kσ)≤ν˜σ(∂u˜σ(S˜σ))=μ˜σ(S˜σ)≤Cμμ˜σ12S˜σ=Cμν˜σ∂u˜σ12S˜σ≤Cμν˜σ(B1rn){\tilde{\nu }}_{\sigma }\left({\pmb{\mathscr{K}}}_{\sigma })\le {\tilde{\nu }}_{\sigma }(\partial {\tilde{u}}_{\sigma }\left({\tilde{\pmb{\mathscr{S}}}}_{\sigma }))={\tilde{\mu }}_{\sigma }\left({\tilde{\pmb{\mathscr{S}}}}_{\sigma })\le {\pmb{\mathscr{C}}}_{\mu }{\tilde{\mu }}_{\sigma }\left(\frac{1}{2}{\tilde{\pmb{\mathscr{S}}}}_{\sigma }\right)={\pmb{\mathscr{C}}}_{\mu }{\tilde{\nu }}_{\sigma }\left(\partial {\tilde{u}}_{\sigma }\left(\frac{1}{2}{\tilde{\pmb{\mathscr{S}}}}_{\sigma }\right)\right)\le {\pmb{\mathscr{C}}}_{\mu }{\tilde{\nu }}_{\sigma }\left({\pmb{\mathscr{B}}}_{\tfrac{1}{{r}_{n}}})(cf. (2.6)), where μ˜σ{\tilde{\mu }}_{\sigma }and ν˜σ{\tilde{\nu }}_{\sigma }are defined so that (∇u˜σ)#μ˜σ=ν˜σ{\left(\nabla {\tilde{u}}_{\sigma })}_{\#}{\tilde{\mu }}_{\sigma }={\tilde{\nu }}_{\sigma }and Kσ≔convBrn∪rn+n(1−τ)2τe1withrn≔12n3/2.{\pmb{\mathscr{K}}}_{\sigma }:= {\rm{conv}}\left({\pmb{\mathscr{B}}}_{{r}_{n}}\cup \left\{\phantom{\rule[-1.25em]{}{0ex}},\frac{{r}_{n}+n\left(1-\tau )}{2\tau }{{\bf{e}}}_{1}\right\}\right)\hspace{1.0em}\hspace{0.1em}\text{with}\hspace{0.1em}\hspace{0.33em}{r}_{n}:= \frac{1}{2{n}^{3\text{/}2}}.This proves that uuis differentiable.By [7, Lemma A.24], for example, we know that differentiable convex functions are continuously differentiable. So we conclude that uuis continuously differentiable in X\pmb{\mathscr{X}}.Part 2. ∇u(X)=Y\nabla u\left(\pmb{\mathscr{X}})=\pmb{\mathscr{Y}}when X\pmb{\mathscr{X}}is convex.Because ∇u\nabla uis continuous in X\pmb{\mathscr{X}}, its image Y′≔∇u(X)\pmb{\mathscr{Y}}^{\prime} := \nabla u\left(\pmb{\mathscr{X}})is an open set of full ν\nu -measure contained inside Y\pmb{\mathscr{Y}}. Also, as the assumptions on μ\mu and ν\nu are symmetric, the optimal transport map ∇v\nabla vfrom ν\nu to μ\mu is continuous, and X′≔∇v(Y)\pmb{\mathscr{X}}^{\prime} := \nabla v\left(\pmb{\mathscr{Y}})is an open set of full μ\mu -measure contained inside X\pmb{\mathscr{X}}. Hence, by recalling that ∇u\nabla uand ∇v\nabla vare inverses of each other [8, Corollary 2.5.13], we conclude that X′=X\pmb{\mathscr{X}}^{\prime} =\pmb{\mathscr{X}}and Y′=Y\pmb{\mathscr{Y}}^{\prime} =\pmb{\mathscr{Y}}, as desired.Part 3. ∇u\nabla uis locally Hölder continuous inside X\pmb{\mathscr{X}}.Thanks to the strict convexity and C1{\pmb{\mathscr{C}}}^{1}regularity of uu, we can localize the arguments of the proof of [11, Theorem 1.1] to obtain the local Hölder continuity of uuinside X\pmb{\mathscr{X}}.More precisely, if u∗{u}^{\ast }denotes the Legendre transform of uu(2.2), as in [11], one can show that u∗{u}^{\ast }satisfies a weak form of Alexandrov’s maximum principle [11, Lemma 3.2], from which one deduces the engulfing property for the sections of u∗{u}^{\ast }[11, Lemma 3.3]. Iteratively applying this engulfing property, one obtains a polynomial strict convexity bound for u∗{u}^{\ast }. This bound implies the local Hölder continuity of uuinside X\pmb{\mathscr{X}}[11, Proof of Theorem 1.1]. We leave the details of this adaptation to the interested reader.

Journal

Advanced Nonlinear Studies – de Gruyter

Published: Jan 1, 2023

Keywords: optimal transport maps; Monge-Ampère equation; doubling measures; unbounded support; 35J96; 49Q22; 52A41

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