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


Authors: Daniel Azagra and Juan Ferrera
Journal: Proc. Amer. Math. Soc. 130 (2002), 3687-3692
MSC (2000): Primary 52A99, 46B99
DOI: https://doi.org/10.1090/S0002-9939-02-06695-9
Published electronically: July 2, 2002
MathSciNet review: 1920049
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Abstract: We show that for every closed convex set $C$ in a separable Banach space $X$ 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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Additional Information

Daniel Azagra
Affiliation: Departamento de Análisis Matemático, Facultad de Ciencias Matemáticas, Universidad Complutense, Madrid, 28040, Spain
Email: Daniel_Azagra@mat.ucm.es

Juan Ferrera
Affiliation: Departamento de Análisis Matemático, Facultad de Ciencias Matemáticas, Universidad Complutense, Madrid, 28040, Spain
Email: ferrera@mat.ucm.es

DOI: https://doi.org/10.1090/S0002-9939-02-06695-9
Received by editor(s): July 9, 2001
Published electronically: July 2, 2002
Communicated by: Jonathan M. Borwein
Article copyright: © Copyright 2002 American Mathematical Society

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