On differentiability of convex operators

The main known results on differentiability of continuous convex operators f from a Banach space X to an ordered Banach space Y are due to J.M. Borwein and N.K. Kirov. Our aim is to prove some “supergeneric” results, i.e., to show that, sometimes, the set of Gâteaux or Fréchet nondifferentiability points is not only a first-category set, but also smaller in a stronger sense. For example, we prove that if Y is countably Daniell and the space L ( X , Y ) of bounded linear operators is separable, then each continuous convex operator f : X → Y is Fréchet differentiable except for a Γ -null angle-small set. Some applications of such supergeneric results are shown.