On Total Convexity, Bregman Projections and Stability in Banach Spaces

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.