Global and fine approximation of convex functions



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Azagra Rueda, Daniel (2013) Global and fine approximation of convex functions. Proceedings of the London Mathematical Society , 107 (Part 4). pp. 799-824. ISSN 0024-6115

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Let U subset of R-d be open and convex. We prove that every (not necessarily Lipschitz or strongly) convex function f:U -> R can be approximated by real analytic convex functions, uniformly on all of U. We also show that C-0-fine approximation of convex functions by smooth (or real analytic) convex functions on R-d is possible in general if and only if d = 1. Nevertheless, for d >= 2, we give a characterization of the class of convex functions on R-d which can be approximated by real analytic (or just smoother) convex functions in the C-0-fine topology. It turns out that the possibility of performing this kind of approximation is not determined by the degree of local convexity or smoothness of the given function, but by its global geometrical behaviour. We also show that every C-1 convex and proper function on U can be approximated by C-infinity convex functions in the C-1-fine topology, and we provide some applications of these results, concerning prescription of (sub-)differential boundary data to convex real analytic functions, and smooth surgery of convex bodies.

Item Type:Article
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"This is a merge of the previous version of this paper with the paper arXiv:1112.1042. This is to be regarded as the final version of those two papers. A slightly different version of this merge will be published in the Proceedings of the London Mathematical Society"

Uncontrolled Keywords:Differential Geometry (math.DG); Classical Analysis and ODEs (math.CA); Functional Analysis (math.FA)
Subjects:Sciences > Mathematics > Functional analysis and Operator theory
ID Code:23643
Deposited On:21 Nov 2013 12:28
Last Modified:07 Feb 2014 11:05

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