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Every closed convex set is the set of minimizers of some C1-smooth convex function

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Azagra Rueda, Daniel and Ferrera Cuesta, Juan (2002) Every closed convex set is the set of minimizers of some C1-smooth convex function. Proceedings of the American Mathematical Society, 130 (12). pp. 3687-3692. ISSN 1088-6826

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Abstract

The authors show that for every closed convex set C in a separable Banach space there is a nonnegative C1 convex function f such that C = {x: f(x) = 0}. The key is to show this for a closed halfspace. This result has several attractive consequences. For example, it provides an easy proof that every closed convex set is the Hausdorff limit of infinitely smooth convex bodies (Cn := {x: f(x) _ 1/n}) and that every continuous convex function is the Mosco limit of C1 convex functions.


Item Type:Article
Subjects:Sciences > Mathematics > Functional analysis and Operator theory
ID Code:12354
Deposited On:07 Mar 2011 12:04
Last Modified:06 Feb 2014 09:23

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