# A Short Proof, Based on Mixed Volumes, of Liggett's Theorem on the Convolution of Ultra-Logconcave Sequences

A Short Proof, Based on Mixed Volumes, of Liggett's Theorem on the Convolution of... R. Pemantle conjectured, and T. M. Liggett proved in 1997, that the convolution of two ultra-logconcave is ultra-logconcave. Liggett's proof is elementary but long. We present here a short proof, based on the mixed volume of convex sets. 1 Introduction Let a = (a ; :::; a ) and b = (b ; :::; b ) be two real sequences. Their convolution c = a ? b 0 m 0 n is de ned as c = a b ; 0  k  n + m. A nonnegative sequence a = (a ; :::; a ) is k i+j=k i j 0 m said to be logconcave if a  a a ; 1  i  m 1: (1) i1 i+1 Following Permantle and [5], we say that a nonnegative sequence a = (a ; :::; a ) is 0 m ultra-logconcave of order d  m (U LC(d)) if the sequence ; 0  i  m is logconcave, ( ) i.e. 0 1 a a a i i1 i+1 @ A ; 1  i  m 1: (2) d d d i i1 i+1 The next result was conjectured by R. Pemantle and proved by T.M. Liggett in 1997 [5]. Theorem 1.1: The convolution of a U LC(l) sequence a and a U LC(d) sequence b is U LC(l + d). Remark 1.2: It is easy to see, by a standard perturbation argument, that it is sucient to consider a positive case: a = (a ; :::; a ); a > 0; 0  i  l and b = (b ; :::; b ); b > 0; 0  i  d: 0 l i 0 d i the electronic journal of combinatorics 16 (2009), #N5 1 The (relatively simple) fact that the convolution of logconcave sequences is also logconcave was proved in [3] in 1949. We present in this paper a short proof of Theorem(1.1). 2 The Minkowski sum and the mixed volume 2.1 The Minkowski sum De nition 2.1: n n 1. Let K ; K  R be two subsets of the Euclidean space R . Their Minkowski sum 1 2 is de ned as K + K = fX + Y : X 2 K ; Y 2 K g: 1 2 1 2 The Minkowski sum is obviously commutative, i.e K + K = K + K , and asso- 1 2 2 1 ciative, i.e K + K + K = K + (K + K ): 1 2 3 1 2 3 l d 2. Let A  R , B  R . Their cartesian product is de ned as l+d A  B := f(X; Y ) 2 R : X 2 A; Y 2 Bg: l+d De ne the next two subsets of R : l+d l+d Lift (A) = f(X; 0) 2 R : X 2 Ag; Lift (B) = f(0; Y ) 2 R : Y 2 Bg: (3) 1 2 Then the next set equalities holds: A  B = Lift (A) + Lift (B): (4) 1 2 The next simple fact will be used below. l d Fact 2.2: Let K ; K  R and C ; C  R . 1 2 1 2 l+d De ne the next two subsets of R : P = K  C ; Q = K  C : 1 1 2 2 Then the following set equality holds: tP + Q = (tK + K )  (tC + C ); t 2 R: (5) 1 2 1 2 Proof: Using (4), we get that (tK + K )  (tC + C ) = Lif t (tK + K ) + Lif t (tC + C ): 1 2 1 2 1 1 2 2 1 2 It follows from the de nition (3) that Lif t (tK + K ) = tLif t (K ) + Lif t (K ); Lif t (tC + C ) = tLif t (C ) + Lif t (C ): 1 1 2 1 1 1 2 2 1 2 2 1 2 2 Therefore, we get by the associativity and commutativity of the Minkowski sum that (tK + K ) (tC + C ) = t(Lif t (K ) + Lif t (C )) + (Lif t (K ) + Lif t (C )) = tP + Q: 1 2 1 2 1 1 2 1 1 2 2 2 the electronic journal of combinatorics 16 (2009), #N5 2 2.2 The mixed volume Let K = (K ; :::; K ) be a n-tuple of convex compact subsets in the Euclidean space 1 n n n R , and let V () be the Euclidean volume in R . It is a well-known result of Herman Minkowski (see for instance [2]), that the functional V ( K +    +  K ) is a ho- n 1 1 n n mogeneous polynomial of degree n with nonnegative coecients, called the Minkowski 00 00 polynomial. Here + denotes Minkowski sum, and K denotes the dilatation of K with coecient   0. The coecient V (K) =: (V (K ; :::; K ) of    : : :   is called the 1 n 1 2 n mixed volume of K ; :::; K . Alternatively, 1 n V (K ; :::; K ) = V ( K +    +  K ); 1 n n 1 1 n n @ :::@ 1 n and X Y V (K ) r ;:::;r 1 n r V ( K +    +  K ) = Q (  ); (6) n 1 1 n n r ! 1in r ++r =n 1in 1 n where the n-tuple K consists of r copies of K ; 1  i  n. r ;:::;r i i 1 n The Alexandrov-Fenchel inequalities [1], [2] state that V (K ; K ; K ; :::; K )  V (K ; K ; K ; :::; K )V (K ; K ; K ; :::; K ): (7) 1 2 3 n 1 1 3 n 2 2 3 n It follows from (6) that if P; Q  R are convex compact sets then V ol (tP + Q) = a t ; t  0; n i 0in 1 1 where a = V ol (Q) = V (Q;    ; Q), a = V (P; Q;    ; Q); : : :, a = V ol (Q) = 0 n 1 n n n! (n1)!1! V (P;    ; P ). n! Using the Alexandrov-Fenchel inequalities (7) we see that the sequence (a ; :::; a ) is 0 n U LC(n). The next remarkable result was proved by G.S. Shephard in 1960: Theorem 2.3: A sequence (a ; :::; a ) is U LC(n) if and only if there exist two convex 0 n compact sets P; Q  R such that a t = V ol (tP + Q); t  0: i n 0in Remark 2.4: The \if" part in Theorem(2.3), which is a particular case of the Alexandrov- Fenchel inequalities, is not simple, but was proved seventy years ago [1]. The proof of the \only if" part in Theorem(2.3) is not dicult and short. G.S. Shephard rst considers the case of positive coecients, which is already sucient for our application. In this positive the electronic journal of combinatorics 16 (2009), #N5 3 P case one chooses Q = f(x ; :::; x ) : x  1; x  0g. In other words, the set Q is 1 n i i 1in the standard simplex in R . And the convex compact set P = Diag( ; :::;  )Q = f(x ; :::; x ) :  1; x  0g;   :::   > 0: 1 n 1 n i 1 n 1jn The general nonnegative case is handled by the topological theory of convex compact subsets. 3 Our proof of Theorem(1.1) Proof: Let a = (a ; :::; a ) be U LC(l) and b = (b ; :::; b ) be U LC(d). De ne two 0 l 0 d P P i j univariate polynomials R (t) = a t and R (t) = a t . 1 i 2 i 0il 0jc Then the polynomial R (t)R (t) := R (t) = c t , where the sequence 1 2 3 0kl+d k c = (c ; :::; c ) is the convolution, c = a ? b. 0 l+d It follows from the \only if"!p part of Theorem(2.3) that R (t) = V ol (tK + K ) and R (t) = V ol (tC + C ); 1 l 1 2 2 d 1 2 l d where K ; K ; C ; C are convex compact sets; K ; K  R and C ; C  R . 1 2 1 2 1 2 1 2 l+d De ne the next two convex compact subsets of R : P = K  C and Q = K  C : 1 1 2 2 l d Here the cartesian product A  B of two subsets A  R and B  R is de ned as l+d A  B := f(X; Y ) 2 R : X 2 A; Y 2 Bg: By Fact(2.2), the Minkowski sum tP + Q = (tK + K )  (tC + C ); t  0. 1 2 1 2 It follows that V ol (tK + K )V ol (tC + C ) = V ol (tP + Q). Therefore the polynomial l 1 2 d 1 2 l+d R (t) = V ol (tP + Q). 3 l+d Finally, we get from the Alexandrov-Fenchel inequalities (the \if" part of Theorem(2.3)) that the sequence of its coecients c = a ? b is U LC(l + d). 4 Final comments 1. Theorem(2.3) and a simple Fact(2.2) allowed us to use very basic (but powerful) representation of the convolution in terms of the product of the corresponding poly- nomials. The original Liggett's proof does not rely on this representation. 2. Let a = (a ; :::; a ) be a real sequence, satisfying the Newton inequalities (2) of 0 m order m. I.e. we dropped the condition of nonnegativity from the de nition of ultra-logconcavity. It is not true that c = a ? a satis es the Newton inequalities of order 2m. the electronic journal of combinatorics 16 (2009), #N5 4 Indeed, consider a = (1; a; 0; b; 1), where a; b > 0 . This real sequence clearly satis es the Newton inequalities of order 4. It follows that c = b ; c = 2a; c = 2(1 ab) and the number 6 5 4 2 2 c a = 2 c c b (1 ab) 4 6 converges to zero if the positive numbers a; b; converge to zero. 3. The reader can nd further implications (and their generalizations) of Theorem(2.3) in [4]. Acknowledgements The author is indebted to the both anonymous reviewers for a careful and thoughtful reading of the original version of this paper. Their corrections and suggestions are re ected in the current version. I would like to thank the U.S. DOE for nancial support through Los Alamos National Laboratory's LDRD program. References [1] A. Aleksandrov, On the theory of mixed volumes of convex bodies, IV, Mixed dis- criminants and mixed volumes (in Russian), Mat. Sb. (N.S.) 3 (1938), 227-251. [2] Yu. D. Burago and V. A. Zalgaller, Geometric Inequalities, Springer-Verlag, 1988. [3] H. Davenport and G. Polya, On the products of two power series, Canad. J. Math. 1(1949), 1-5. [4] L. Gurvits, On multivariate Newton(like) inequalities, available at http://arxiv.org/abs/0812.3687, 2008. [5] T. M. Ligggett, Ultra Logconcave sequences and Negative dependence, Journal of Combinatorial Theory, Series A 79, 315-325, 1997. [6] G. C. Shephard, Inequalities between mixed volumes of convex sets, Mathematika 7 (1960) , 125-138. the electronic journal of combinatorics 16 (2009), #N5 5 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png The Electronic Journal of Combinatorics Unpaywall

# A Short Proof, Based on Mixed Volumes, of Liggett's Theorem on the Convolution of Ultra-Logconcave Sequences

The Electronic Journal of CombinatoricsFeb 13, 2009
5 pages

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### Abstract

R. Pemantle conjectured, and T. M. Liggett proved in 1997, that the convolution of two ultra-logconcave is ultra-logconcave. Liggett's proof is elementary but long. We present here a short proof, based on the mixed volume of convex sets. 1 Introduction Let a = (a ; :::; a ) and b = (b ; :::; b ) be two real sequences. Their convolution c = a ? b 0 m 0 n is de ned as c = a b ; 0  k  n + m. A nonnegative sequence a = (a ; :::; a ) is k i+j=k i j 0 m said to be logconcave if a  a a ; 1  i  m 1: (1) i1 i+1 Following Permantle and [5], we say that a nonnegative sequence a = (a ; :::; a ) is 0 m ultra-logconcave of order d  m (U LC(d)) if the sequence ; 0  i  m is logconcave, ( ) i.e. 0 1 a a a i i1 i+1 @ A ; 1  i  m 1: (2) d d d i i1 i+1 The next result was conjectured by R. Pemantle and proved by T.M. Liggett in 1997 [5]. Theorem 1.1: The convolution of a U LC(l) sequence a and a U LC(d) sequence b is U LC(l + d). Remark 1.2: It is easy to see, by a standard perturbation argument, that it is sucient to consider a positive case: a = (a ; :::; a ); a > 0; 0  i  l and b = (b ; :::; b ); b > 0; 0  i  d: 0 l i 0 d i the electronic journal of combinatorics 16 (2009), #N5 1 The (relatively simple) fact that the convolution of logconcave sequences is also logconcave was proved in [3] in 1949. We present in this paper a short proof of Theorem(1.1). 2 The Minkowski sum and the mixed volume 2.1 The Minkowski sum De nition 2.1: n n 1. Let K ; K  R be two subsets of the Euclidean space R . Their Minkowski sum 1 2 is de ned as K + K = fX + Y : X 2 K ; Y 2 K g: 1 2 1 2 The Minkowski sum is obviously commutative, i.e K + K = K + K , and asso- 1 2 2 1 ciative, i.e K + K + K = K + (K + K ): 1 2 3 1 2 3 l d 2. Let A  R , B  R . Their cartesian product is de ned as l+d A  B := f(X; Y ) 2 R : X 2 A; Y 2 Bg: l+d De ne the next two subsets of R : l+d l+d Lift (A) = f(X; 0) 2 R : X 2 Ag; Lift (B) = f(0; Y ) 2 R : Y 2 Bg: (3) 1 2 Then the next set equalities holds: A  B = Lift (A) + Lift (B): (4) 1 2 The next simple fact will be used below. l d Fact 2.2: Let K ; K  R and C ; C  R . 1 2 1 2 l+d De ne the next two subsets of R : P = K  C ; Q = K  C : 1 1 2 2 Then the following set equality holds: tP + Q = (tK + K )  (tC + C ); t 2 R: (5) 1 2 1 2 Proof: Using (4), we get that (tK + K )  (tC + C ) = Lif t (tK + K ) + Lif t (tC + C ): 1 2 1 2 1 1 2 2 1 2 It follows from the de nition (3) that Lif t (tK + K ) = tLif t (K ) + Lif t (K ); Lif t (tC + C ) = tLif t (C ) + Lif t (C ): 1 1 2 1 1 1 2 2 1 2 2 1 2 2 Therefore, we get by the associativity and commutativity of the Minkowski sum that (tK + K ) (tC + C ) = t(Lif t (K ) + Lif t (C )) + (Lif t (K ) + Lif t (C )) = tP + Q: 1 2 1 2 1 1 2 1 1 2 2 2 the electronic journal of combinatorics 16 (2009), #N5 2 2.2 The mixed volume Let K = (K ; :::; K ) be a n-tuple of convex compact subsets in the Euclidean space 1 n n n R , and let V () be the Euclidean volume in R . It is a well-known result of Herman Minkowski (see for instance [2]), that the functional V ( K +    +  K ) is a ho- n 1 1 n n mogeneous polynomial of degree n with nonnegative coecients, called the Minkowski 00 00 polynomial. Here + denotes Minkowski sum, and K denotes the dilatation of K with coecient   0. The coecient V (K) =: (V (K ; :::; K ) of    : : :   is called the 1 n 1 2 n mixed volume of K ; :::; K . Alternatively, 1 n V (K ; :::; K ) = V ( K +    +  K ); 1 n n 1 1 n n @ :::@ 1 n and X Y V (K ) r ;:::;r 1 n r V ( K +    +  K ) = Q (  ); (6) n 1 1 n n r ! 1in r ++r =n 1in 1 n where the n-tuple K consists of r copies of K ; 1  i  n. r ;:::;r i i 1 n The Alexandrov-Fenchel inequalities [1], [2] state that V (K ; K ; K ; :::; K )  V (K ; K ; K ; :::; K )V (K ; K ; K ; :::; K ): (7) 1 2 3 n 1 1 3 n 2 2 3 n It follows from (6) that if P; Q  R are convex compact sets then V ol (tP + Q) = a t ; t  0; n i 0in 1 1 where a = V ol (Q) = V (Q;    ; Q), a = V (P; Q;    ; Q); : : :, a = V ol (Q) = 0 n 1 n n n! (n1)!1! V (P;    ; P ). n! Using the Alexandrov-Fenchel inequalities (7) we see that the sequence (a ; :::; a ) is 0 n U LC(n). The next remarkable result was proved by G.S. Shephard in 1960: Theorem 2.3: A sequence (a ; :::; a ) is U LC(n) if and only if there exist two convex 0 n compact sets P; Q  R such that a t = V ol (tP + Q); t  0: i n 0in Remark 2.4: The \if" part in Theorem(2.3), which is a particular case of the Alexandrov- Fenchel inequalities, is not simple, but was proved seventy years ago [1]. The proof of the \only if" part in Theorem(2.3) is not dicult and short. G.S. Shephard rst considers the case of positive coecients, which is already sucient for our application. In this positive the electronic journal of combinatorics 16 (2009), #N5 3 P case one chooses Q = f(x ; :::; x ) : x  1; x  0g. In other words, the set Q is 1 n i i 1in the standard simplex in R . And the convex compact set P = Diag( ; :::;  )Q = f(x ; :::; x ) :  1; x  0g;   :::   > 0: 1 n 1 n i 1 n 1jn The general nonnegative case is handled by the topological theory of convex compact subsets. 3 Our proof of Theorem(1.1) Proof: Let a = (a ; :::; a ) be U LC(l) and b = (b ; :::; b ) be U LC(d). De ne two 0 l 0 d P P i j univariate polynomials R (t) = a t and R (t) = a t . 1 i 2 i 0il 0jc Then the polynomial R (t)R (t) := R (t) = c t , where the sequence 1 2 3 0kl+d k c = (c ; :::; c ) is the convolution, c = a ? b. 0 l+d It follows from the \only if"!p part of Theorem(2.3) that R (t) = V ol (tK + K ) and R (t) = V ol (tC + C ); 1 l 1 2 2 d 1 2 l d where K ; K ; C ; C are convex compact sets; K ; K  R and C ; C  R . 1 2 1 2 1 2 1 2 l+d De ne the next two convex compact subsets of R : P = K  C and Q = K  C : 1 1 2 2 l d Here the cartesian product A  B of two subsets A  R and B  R is de ned as l+d A  B := f(X; Y ) 2 R : X 2 A; Y 2 Bg: By Fact(2.2), the Minkowski sum tP + Q = (tK + K )  (tC + C ); t  0. 1 2 1 2 It follows that V ol (tK + K )V ol (tC + C ) = V ol (tP + Q). Therefore the polynomial l 1 2 d 1 2 l+d R (t) = V ol (tP + Q). 3 l+d Finally, we get from the Alexandrov-Fenchel inequalities (the \if" part of Theorem(2.3)) that the sequence of its coecients c = a ? b is U LC(l + d). 4 Final comments 1. Theorem(2.3) and a simple Fact(2.2) allowed us to use very basic (but powerful) representation of the convolution in terms of the product of the corresponding poly- nomials. The original Liggett's proof does not rely on this representation. 2. Let a = (a ; :::; a ) be a real sequence, satisfying the Newton inequalities (2) of 0 m order m. I.e. we dropped the condition of nonnegativity from the de nition of ultra-logconcavity. It is not true that c = a ? a satis es the Newton inequalities of order 2m. the electronic journal of combinatorics 16 (2009), #N5 4 Indeed, consider a = (1; a; 0; b; 1), where a; b > 0 . This real sequence clearly satis es the Newton inequalities of order 4. It follows that c = b ; c = 2a; c = 2(1 ab) and the number 6 5 4 2 2 c a = 2 c c b (1 ab) 4 6 converges to zero if the positive numbers a; b; converge to zero. 3. The reader can nd further implications (and their generalizations) of Theorem(2.3) in [4]. Acknowledgements The author is indebted to the both anonymous reviewers for a careful and thoughtful reading of the original version of this paper. Their corrections and suggestions are re ected in the current version. I would like to thank the U.S. DOE for nancial support through Los Alamos National Laboratory's LDRD program. References [1] A. Aleksandrov, On the theory of mixed volumes of convex bodies, IV, Mixed dis- criminants and mixed volumes (in Russian), Mat. Sb. (N.S.) 3 (1938), 227-251. [2] Yu. D. Burago and V. A. Zalgaller, Geometric Inequalities, Springer-Verlag, 1988. [3] H. Davenport and G. Polya, On the products of two power series, Canad. J. Math. 1(1949), 1-5. [4] L. Gurvits, On multivariate Newton(like) inequalities, available at http://arxiv.org/abs/0812.3687, 2008. [5] T. M. Ligggett, Ultra Logconcave sequences and Negative dependence, Journal of Combinatorial Theory, Series A 79, 315-325, 1997. [6] G. C. Shephard, Inequalities between mixed volumes of convex sets, Mathematika 7 (1960) , 125-138. the electronic journal of combinatorics 16 (2009), #N5 5

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The Electronic Journal of CombinatoricsUnpaywall

Published: Feb 13, 2009