Abstract

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∈(1,+∞)r\in(1,+\infty) 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.

Author information

Contact details are reproduced from the original publication and may be historical.

Elena Resmerita

Dept. of Mathematics, University of Haifa, 31905 Haifa, Israel

Keywords

  • Bregman distance
  • Bregman projection
  • total convexity
  • essential strict convexity
  • E-space
  • Mosco convergence

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Suggested citation

E. Resmerita. “On Total Convexity, Bregman Projections and Stability in Banach Spaces.” Journal of Convex Analysis 11 (2004), No. 1, 1–16. https://doi.org/10.68381/jca11001

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Published by Heldermann Verlag, 2004. Rights now held by Banach Press LLC.