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

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. c 2010 Published by Elsevier Inc.

• Type: Journal
• Year: 2010
• Where: JAT
• Authors: Chong Li, Wen Song, Jen-Chih Yao
• Keywords: Banach Space X, JAT 2010, Projection Operator, Sufficient Conditions