This is an announcement for the paper "Global approximation of convex functions" by D. Azagra.

Abstract: We show that for every (not necessarily bounded) open convex subset $U$ of $\R^n$, every (not necessarily Lipschitz or strongly) convex function $f:U\to\R$ can be approximated by real analytic convex functions, uniformly on all of $U$. In doing so we provide a technique which transfers results on uniform approximation on bounded sets to results on uniform approximation on unbounded sets, in such a way that not only convexity and $C^k$ smoothness, but also local Lipschitz constants, minimizers, order, and strict or strong convexity, are preserved. This transfer method is quite general and it can also be used to obtain new results on approximation of convex functions defined on Riemannian manifolds or Banach spaces. We also provide a characterization of the class of convex functions which can be uniformly approximated on $\R^n$ by strongly convex functions.

Archive classification: math.FA math.CA math.DG

Mathematics Subject Classification: 26B25, 41A30, 52A1, 46B20, 49N99, 58E99

Remarks: 16 pages

Submitted from: dazagra@gmail.com

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1112.1042

or

http://arXiv.org/abs/1112.1042