Every closed convex set is the set of minimizers of some $C^{\infty }$-smooth convex function
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- by Daniel Azagra and Juan Ferrera
- Proc. Amer. Math. Soc. 130 (2002), 3687-3692
- DOI: https://doi.org/10.1090/S0002-9939-02-06695-9
- Published electronically: July 2, 2002
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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.References
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Bibliographic 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
- Received by editor(s): July 9, 2001
- Published electronically: July 2, 2002
- Communicated by: Jonathan M. Borwein
- © Copyright 2002 American Mathematical Society
- 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
- MathSciNet review: 1920049