On a Conjecture of E. Dittert

AbstractLet Kn denote the set of all n X n nonnegative matrices whose entries have sum n, and let φ be a real valued function defined on Kn by φ(X) = πin=1 n, + πj=1cj−n per X for X E Kn with row sum vector (r1,…, rn) and column sum vector (cl,hellip;, cn). For the same X, let φij(X)= πk≠irk + π1≠jc1 - per X(i| j). A ϵKn is called a φ-maximizing matrix if φ(A) > φ(X) for all X ϵ Kn. Dittert's conjecture asserts that Jn = [1/n]n×n is the unique (φ-maximizing matrix on Kn. In this paper, the following are proved: (i) If A = [aij] is a φ-maximizing matrix on Kn then φij(A) = φ (A) if aij > 0, and φij (A) ⩽ φ (A)if aij = 0. (ii) The conjecture is true for n = 3