Convexity of generalized proximinal sets in Banach spaces

For a bounded closed convex subset C of a Banach space X with the origin as an interior point, we study the convexity issue of proximinal sets G in Banach spaces in the sense of generalized best approximation determined by the Minkowski functional generated by C. Some sufficient conditions such as the smoothness and compactly locally uniform convexity of C, or the weakly* uniform convexity of C. are provided for a proximinal set G to be convex.