Maximization of a matrix function related to the Dittert conjecture

AbstractLet Kn denote the set of all nonnegative n×n matrices whose entries have sum n, and let Jn=[1n]n×n. For k = 1,…, n, and for A ∈ Kn with row sum s r1,…, rn and column sums c1,…, cn, let ϕk be defined by ϕk(A) = ∑α, β∈Qk∏i ∈ α ri + ∏j ∈ β cj − per A[α|β]. We propose a problem of maximizing ϕk of which the Dittert conjecture is a special case, and obtain some results related to this problem. Many of known theorems for ϕn are shown to hold for ϕk, k=1,…, n