Maximization of a matrix function related to the Dittert conjecture

Abstract Let K n denote the set of all nonnegative n × n matrices whose entries have sum n , and let J n =[ 1 n ] n×n . For k = 1,…, n , and for A ∈ K n with row sum s r 1 ,…, r n and column sums c 1 ,…, c n , let ϕ k be defined by ϕ k (A) = ∑ α, β∈ Q k ∏ i ∈ α r i + ∏ j ∈ β c j − per A[α|β] . We propose a problem of maximizing ϕ k of which the Dittert conjecture is a special case, and obtain some results related to this problem. Many of known theorems for ϕ n are shown to hold for ϕ k , k =1,…, n .