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Journal of Convex Analysis 11 (2004), No. 1, 001--016
Copyright Heldermann Verlag 2004
On Total Convexity, Bregman Projections and Stability in Banach Spaces
Elena Resmerita
Dept. of Mathematics, University of Haifa, 31905 Haifa, Israel
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.
Keywords: Bregman distance, Bregman projection, total convexity, essential strict
convexity, E-space, Mosco convergence.
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