Proceedings of the American Mathematical Society

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ISSN 1088-6826(e) ISSN 0002-9939(p)

Every closed convex set is the set of minimizers of some $C^{\infty}$ -smooth convex function

Author(s): Daniel Azagra; Juan Ferrera
Journal: Proc. Amer. Math. Soc. 130 (2002), 3687-3692.
MSC (2000): Primary 52A99, 46B99
Posted: July 2, 2002
MathSciNet review: 1920049
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Abstract: We show that for every closed convex set in a separable Banach space there is a $C^{\infty}$ -smooth convex function $f:X\longrightarrow [0,\infty)$ 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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