Institute of Mathematics and Informatics Bulgarian Academy of Sciences
Serdica Mathematical Journal, Vol. 22, No 3, (1996), 307p-330p
To a convex set in a Banach space we associate a convex function
(the separating function), whose subdifferential provides useful information on the
nature of the supporting and exposed points of the convex set. These points are
shown to be also connected to the solutions of a minimization problem involving the
separating function. We investigate some relevant properties of this function and of
its conjugate in the sense of Legendre-Fenchel. Then we highlight the connections
between set convergence, with respect to the slice and Attouch-Wets topologies,
and convergence, in the same sense, of the associated functions. Finally, by using
known results on the behaviour of the subdifferential of a convex function under
the former epigraphical perturbations, we are able to derive stability results for
the set of supported points and of supporting and exposing functionals of a closed
convex subset of a Banach space.
* This work was supported by the CNR while the author was visiting the University of Milan.