The Bregman distance, approximate compactness and convexity of Chebyshev sets in Banach spaces

Abstract

We present some sufficient conditions ensuring the upper semicontinuity and the continuity of the Bregman projection operator Π g C and the relative projection operator P g C in terms of the D-approximate (weak) compactness for a nonempty closed set C in a Banach space X . We next present certain sufficient conditions as well as equivalent conditions for the convexity of a Chebyshev subset of a Banach space X . Our results extend the corresponding results of [H.H. Bauschke, X.F. Wang, J. Ye and X. M. Yuan, Bregman distances and Chebyshev sets, J. Approx. Theory 159 (2009) 3-25] to infinite dimensional spaces.

A characterization of essentially strictly convex functions on reflexive Banach spaces

“…Remark 2. Since there are Legendre functions that are cofinite but not 1coercive ([BBC, Example 7.5]), Corollary 1 above improves a bit [LSY,Theorem 4.1].…”

Prediction in Riemannian metrics derived from divergence functions

“…Fisher [16]. It plays a role in optimization theory as in Baushke and Borwein [5], Baushke and Lewis [6], Baushke and Combettes [8], Censor and Reich [13], Baushke et al [7] and Censor and Zaknoon [14], or to solve operator equations as in Butnariu and Resmerita [11], in approximation theory in Banach spaces as in Baushke and Combettes [8] or Li et al [19]. In applications of geometry to statistics and information theory as in Amari and Nagaoka [2], Csiszár [15], Amari and Cichoski [1], Calin and Urdiste [12] or Nielsen [22].…”

One Property of Bregman Projection onto Closed Convex Sets in Banach Spaces and Applications

“…For the more details and other related results, the readers are referred to (L.M. Bregman, 1967, p200-217), (D. Butnariu, E. Resmerita, 2006, p1-39), (C. Li, W. Song, J.C. Yao, 2010, p1128-1149.…”