On differentiability of convex operators (Q1947258)

Let \(X\) be a Banach space and \(Y\) be an ordered normed space. Let \(A \subset X\) be an open convex set. A mapping \( f:A\rightarrow Y\) is called a convex operator if \(f(\lambda_1 x_1 + \lambda_2 x_2) \leq \lambda_1f(x_1) + \lambda_2 f(x_2)\) for \(x_1, x_2 \in A\) and \(0\leq \lambda_1, \lambda_2\), \(\lambda_1 + \lambda_2 = 1\). The aim of the paper under review is to prove that the set of Gâteaux or Fréchet nondifferentiability points of a continuous convex operator is not only a first-category set, but is also small in a strong sense. The main result of the paper is the following theorem. Theorem. Let \(A\) be an open convex set in the Banach space \(X\) and \(Y\) be an ordered Banach space with the property that each decreasing sequence of positive elements in \(Y\) is convergent. If the space \(L(X,Y)\) of bounded linear operators is separable, then each continuous convex operator \(T:A\rightarrow Y\) is Fréchet differentiable except for a \(\Gamma\)-null angle-small set.