Geodesic
On Total Convexity, Bregman Projections and Stability in Banach Spaces
Journal of convex analysis, Tome 11 (2004) no. 1, pp. 001-016.

Voir la notice de l'article provenant de la source Heldermann Verlag

Totally convex functions and Bregman projections associated to them are of special interest for building optimization and feasibility algorithms. This motivates one to investigate existence of totally convex functions in Banach spaces. Also, this raises the question whether and under which conditions the corresponding Bregman projections have the properties needed for guaranteeing convergence and stability of the algorithms based on them. We show that a reflexive Banach space in which some power r greater than 1 of the norm is totally convex is an E-space and conversely. Also we prove that totally convex functions in reflexive Banach spaces are necessarily essentially strictly convex in the sense of H. H. Bauschke, J. M. Borwein, and P. L. Combettes ["Essential smoothness, essential strict convexity, and Legendre functions in Banach spaces", Commun. Contemp. Math. 3(4) (2001) 615--647]. We use these facts in order to establish continuity and stability properties of Bregman projections.
Mots-clés : Bregman distance, Bregman projection, total convexity, essential strict convexity, E-space, Mosco convergence 
@article{JCA_2004_11_1_JCA_2004_11_1_a0,
     author = {E. Resmerita},
     title = {On {Total} {Convexity,} {Bregman} {Projections} and {Stability} in {Banach} {Spaces}},
     journal = {Journal of convex analysis},
     pages = {001--016},
     publisher = {mathdoc},
     volume = {11},
     number = {1},
     year = {2004},
     url = {http://geodesic.mathdoc.fr/item/JCA_2004_11_1_JCA_2004_11_1_a0/}
}
TY  - JOUR
AU  - E. Resmerita
TI  - On Total Convexity, Bregman Projections and Stability in Banach Spaces
JO  - Journal of convex analysis
PY  - 2004
SP  - 001
EP  - 016
VL  - 11
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/JCA_2004_11_1_JCA_2004_11_1_a0/
ID  - JCA_2004_11_1_JCA_2004_11_1_a0
ER  - 
%0 Journal Article
%A E. Resmerita
%T On Total Convexity, Bregman Projections and Stability in Banach Spaces
%J Journal of convex analysis
%D 2004
%P 001-016
%V 11
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/JCA_2004_11_1_JCA_2004_11_1_a0/
%F JCA_2004_11_1_JCA_2004_11_1_a0
E. Resmerita. On Total Convexity, Bregman Projections and Stability in Banach Spaces. Journal of convex analysis, Tome 11 (2004) no. 1, pp. 001-016. http://geodesic.mathdoc.fr/item/JCA_2004_11_1_JCA_2004_11_1_a0/