Brunn-Minkowski Inequalities for Contingency Tables and Integer Flows
Combinatorics
2007-05-23 v1 Metric Geometry
Abstract
Given a non-negative mxn matrix W=(w_ij) and positive integer vectors R=(r_1, >..., r_m) and C=(c_1, ..., c_n), we consider the total weight T(R, C; W) of mxn non-negative integer matrices (contingency tables) D with the row sums r_i, the column sums c_j, and the weight of D=(d_ij) equal to product of w_ij^d_ij. 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(R_k, C_k; W), where (R, C) is a convex combination of (R_k, C_k) for k=1, ..., p.
Cite
@article{arxiv.math/0603655,
title = {Brunn-Minkowski Inequalities for Contingency Tables and Integer Flows},
author = {Alexander Barvinok},
journal= {arXiv preprint arXiv:math/0603655},
year = {2007}
}
Comments
16 pages