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

Azé, 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 -