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 ΠCg and the relative projection operator PCg in terms of the DD-approximate (weak) compactness for a nonempty closed set CC in a Banach space XX. We next present certain sufficient conditions as well as equivalent conditions for the convexity of a Chebyshev subset of a Banach space XX. 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.