BREGMAN DISTANCES AND KLEE SETS IN BANACH SPACES

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.