Differentiable approximation and extension of convex functions

Abstract

The class of convex functions and, in particular, the class of differentiable convex functions play a very important role in the field of Mathematical Analysis and they have plenty of applications in other disciplines such as Differential Geometry, PDE theory (for instance, Monge-Ampère equations), Non linear Dynamics, Quantum Computing or Economics. Therefore, it is no doubt useful to be able to approximate or to extend by differentiable convex functions in various Banach spaces. If we are given a convex function bounded on bounded sets defined on a Banach space whose dual norm is LUR, recent results by D. Azagra ensure that this function can be approximated by differentiable convex functions uniformly on the whole space. However, since there are example of continuous convex functions which are not bounded on bounded subsets, it is desirable to improve these results in such away that this restriction on the function to be approximated can be removed. In this thesis, we drop this assumption and show that every continuous convex function (not necessarily bounded on bounded subsets) defined on an open convex subset of a Banach space whose dual norm is LUR, can be uniformly approximated by differentiable convex functions. This result is a consequence of a more general result which shows that the problem of approximating continuous convex functions uniformly by convex functions of a certain differentiability class can be reduced to the case when the original functions are, in addition, Lipschitz...