Formula omitted]-subdifferential of Jensen-convex functions

Abstract

A real function is called radially Q -differentiable at the point x if, for every real number h, the finite limit d Q f ( x , h ) of the ratio ( f ( x + r h ) − f ( x ) ) / r exists whenever r tends to zero through the positive rationals. We establish that, in particular, Jensen-convex functions are everywhere radially Q -differentiable. Moreover, if f is Jensen-convex, then, for each x, the mapping h ↦ d Q f ( x , h ) is subadditive, and it is an upper bound for any additive mapping A satisfying the inequality f ( x ) + A ( y − x ) ⩽ f ( y ) for every y. We also characterize all set-valued mappings built up from additive solutions A of this inequality with some Jensen-convex function f.