@VictorKleptsyn Yes, you're right. Essentially we want to maximize a convex function over a convex set, thus the maximum would be obtained at an extreme point, and in two dimensional case, this is the permutation matrix. However, since maximizing a convex function is generally hard, we cannot expect a solution for arbitrary distribution $\nu$, but I hope for some special $\nu$ such as the uniform distribution, we can obtain some result.