# Stability of Supporting and Exposing Elements of Convex Sets in Banach Spaces

Serdica Mathematical Journal (1996)

- Volume: 22, Issue: 3, page 307-330
- ISSN: 1310-6600

## Access Full Article

top## Abstract

top## How to cite

topAzé, D., and Lucchetti, R.. "Stability of Supporting and Exposing Elements of Convex Sets in Banach Spaces." Serdica Mathematical Journal 22.3 (1996): 307-330. <http://eudml.org/doc/11639>.

@article{Azé1996,

abstract = {* This work was supported by the CNR while the author was visiting the University of Milan.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.},

author = {Azé, D., Lucchetti, R.},

journal = {Serdica Mathematical Journal},

keywords = {Convex Sets; Convex Functions; Supported and Exposed Points; Slice Topology; Attouch-Wets Topology; Convex Optimization; convex set; convex function; separating function; subdifferential; supporting and exposed points; conjugate in the sense of Legendre-Fenchel; Attouch-Wets topologies},

language = {eng},

number = {3},

pages = {307-330},

publisher = {Institute of Mathematics and Informatics Bulgarian Academy of Sciences},

title = {Stability of Supporting and Exposing Elements of Convex Sets in Banach Spaces},

url = {http://eudml.org/doc/11639},

volume = {22},

year = {1996},

}

TY - JOUR

AU - Azé, D.

AU - Lucchetti, R.

TI - Stability of Supporting and Exposing Elements of Convex Sets in Banach Spaces

JO - Serdica Mathematical Journal

PY - 1996

PB - Institute of Mathematics and Informatics Bulgarian Academy of Sciences

VL - 22

IS - 3

SP - 307

EP - 330

AB - * This work was supported by the CNR while the author was visiting the University of Milan.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.

LA - eng

KW - Convex Sets; Convex Functions; Supported and Exposed Points; Slice Topology; Attouch-Wets Topology; Convex Optimization; convex set; convex function; separating function; subdifferential; supporting and exposed points; conjugate in the sense of Legendre-Fenchel; Attouch-Wets topologies

UR - http://eudml.org/doc/11639

ER -

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.