Alexander Barvinok

Abstract:   We discuss a certain approximate log-concavity property for a wide family of combinatorially defined quantities. Examples include the number of non-negative integer matrices (contingency tables) with prescribed row and column sums (margins) as a function of the margins and the number of integer feasible flows in a network, as a function of the excesses at the vertices. Geometrically, we are talking about a version of the Brunn-Minkowski inequality for the number of integer points in a class of polytopes, known as transportation polytopes. This approximate log-concavity results in asymptotic log-concavity under certain natural scaling of parameters and some of the quantities may even be genuinely log-concave. We speculate on possible relations to approximate and genuine log-concavity of constants coming from the representation theory.

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