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Every Closed Convex Set Is the Set of Minimizers of Some C-Smooth Convex Function

Daniel Azagra and Juan Ferrera
Proceedings of the American Mathematical Society
Vol. 130, No. 12 (Dec., 2002), pp. 3687-3692
Stable URL: http://www.jstor.org/stable/1194411
Page Count: 6
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Every Closed Convex Set Is the Set of Minimizers of Some C∞-Smooth Convex Function
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Abstract

We show that for every closed convex set C in a separable Banach space X there is a C-smooth "convex" function f: X→ [0,∞) so that f-1(0)=C. We also deduce some interesting consequences concerning smooth approximation of closed convex sets and continuous convex functions.

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