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PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY
 

Summary: PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 130, Number 12, Pages 3687­3692
S 0002-9939(02)06695-9
Article electronically published on July 2, 2002
EVERY CLOSED CONVEX SET IS THE SET OF MINIMIZERS
OF SOME C
-SMOOTH CONVEX FUNCTION
DANIEL AZAGRA AND JUAN FERRERA
(Communicated by Jonathan M. Borwein)
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.
It is well known that if a separable Banach space has a C1
-smooth equivalent
norm, then every closed convex C set can be regarded as the set of minimizers of a
C1
-smooth convex function f. One can obtain such a function f by considering the
inf-convolution of the smooth norm with the indicator function of C (valued 0 on C

  

Source: Azagra Rueda, Daniel - Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid

 

Collections: Mathematics