  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.
