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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. 799824. ISSN 00246115

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Official URL: http://plms.oxfordjournals.org/content/107/4/799
URL  URL Type 

http://www.oxfordjournals.org/  Publisher 
http://arxiv.org/abs/1201.4760  Organisation 
Abstract
Let U subset of Rd 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 C0fine approximation of convex functions by smooth (or real analytic) convex functions on Rd 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 Rd which can be approximated by real analytic (or just smoother) convex functions in the C0fine 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 C1 convex and proper function on U can be approximated by Cinfinity convex functions in the C1fine 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 

Additional Information:  "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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