Brunn-Minkowski Inequalities for Contingency Tables and Integer Flows
Abstract
Given a non-negative mxn matrix W=(wij) and positive integer vectors R=(r1, >..., rm) and C=(c1, ..., cn), we consider the total weight T(R, C; W) of mxn non-negative integer matrices (contingency tables) D with the row sums ri, the column sums cj, and the weight of D=(dij) equal to product of wijdij. In particular, if W is a 0-1 matrix, T(R, C; W) is the number of integer feasible flows in a bipartite network. We prove a version of the Brunn-Minkowski inequality relating the numbers T(R, C; W) and T(Rk, Ck; W), where (R, C) is a convex combination of (Rk, Ck) for k=1, ..., p.
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