 
Summary: PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 130, Number 12, Pages 36873692
S 00029939(02)066959
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 Csmooth convex function f : X  [0, ) so that
f1(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
infconvolution of the smooth norm with the indicator function of C (valued 0 on C
