The generic differentiability of convex-concave functions: Characterization

Abstract

As established by R T. Rockafellar, real valued convex-concave functions are generically differentiable. It this paper we shall show that for a convex-concave function defined on an open convex set $C \times D,$ there exist dense subsets ${\cal N}$ of $C$ and ${\cal M}$ of $D$ such that the partial derivative with respect to the first variable (resp. second variable) exists on ${\cal N} \times D$ (resp. $C \times {\cal M}$) and therefore the function is differentiable on ${\cal N} \times {\cal M}$. This is an interesting property of convex-concave functions and it does not hold for convex-convex functions. As an immediate application we recover the generic single-valuedness of monotone operators.