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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
Published by: American Mathematical Society
Stable URL: http://www.jstor.org/stable/1194411
Page Count: 6
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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.
Proceedings of the American Mathematical Society © 2002 American Mathematical Society