BREGMAN DISTANCES AND KLEE SETS IN BANACH SPACES

Abstract

In this paper, we first present some sufficient conditions for the upper semicontinuity and/or the continuity of the Bregman farthest-point map $Q_C^g$ and the relative farthest-point map $S_C^g$ for a nonempty $D$-maximally approximately compact subset $C$ of a Banach space $X$. We next present certain sufficient conditions as well as equivalent conditions for a Klee set to be singleton in a Banach space $X$. Our results extend and/or improve the corresponding ones of [Bauschke, et al., J. Approx. Theory, 158 (2009), pp. 170-183] to infinite dimensional spaces.