A note on the convexity of the sum of subpermanents Original Research Article

Let Ωn denote the set of all n × n doubly stochastic matrices, and let σk(A) be the sum of all subpermanents of order k of matrix A. We prove that σ2(A) and σ3(A) are convex on Ωn for n ≥ 2 and n ≥ 4, respectively, and also conjecture the following: For every k ≥ 3 there exists nk ≥ k + 1 such that the inequality σk (αJn + (1 − α)A) ≤ ασk(Jn) + (1 − α)σk(A) holds for all α set membership, variant [0, 1] and all A set membership, variant Ωn with n ≥ nk, where Jn = (1/n)ni,j=1 set membership, variant Ωn. It is shown that this conjecture is true for k ≤ 4 with n3 = 4 and n4 = 6.