Differentiability of convex functions on sublinear topological spaces and variational principles in locally convex spaces

This paper presents a type of variational principles for real valued w* lower semicontinuous functions on certain subsets in duals of locally convex spaces, and resolve a problem concerning differentiability of convex functions on general Banach spaces. They are done through discussing differentiability of convex functions on nonlinear topological spaces and convexification of nonconvex functions on topological linear spaces.