Characterizations of uniform convexity for differentiable functions

Abstract

We consider convex functions in d real variables. For applications, for example in optimization, various strengthened forms of convexity have been introduced. Among them, uniform convexity is one of the most general, defined by involving a so-called modulus ?. Inspired by three classical characterizations of ordinary convexity, we aim at characterizations of uniform convexity by conditions in terms of the gradient or the Hessian matrix of the considered function for certain classes of moduli ?.