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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 sucient 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 coecients, called the Minkowski 00 00 polynomial. Here + denotes Minkowski sum, and K denotes the dilatation of K with coecient 0. The coecient 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 ! 1in r ++r =n 1in 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 0in 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 0in 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 dicult and short. G.S. Shephard rst considers the case of positive coecients, which is already sucient 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 1in 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 1jn 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 0il 0jc Then the polynomial R (t)R (t) := R (t) = c t , where the sequence 1 2 3 0kl+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 coecients 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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Published: Feb 13, 2009
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