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In a real Banach space X, we introduce for a non-empty set C in X the notion of suns in the sense of Bregman distances and show that C is such a sun if and only if C is convex. Also, we giv...

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 are provided for a proximinal set G to be convex.

Let X be a real Banach space, C a closed bounded convex subset of X with the origin as an interior point, and $$p_C$$ the Minkowski functional generated by the set C. This paper is concerned with the problem of generalized best approximation with respect to $$p_C$$ . A property $$(\varepsilon _*)$$ concerning a subspace of $$X^*$$ is introduced to characterize generalized proximinal subspaces in X. A set C with feature as above in the space $$l_1$$ of absolutely summable sequences of real numbers and a continuous linear functional f on $$l_1$$ are constructed to show that each point in an open half space determined by the kernel of f admits a generalized best approximation from the kernel but each point in the other open half space does not.

The notions of the τ C -Kolmogorov condition, the τ C -sun and the τ C -regular point are introduced, and the relationships between them and the best τ C -approximation are explored. As a consequence, characterizations of best τ C -approximations from some kind of subsets (not necessarily convex) are obtained. As an application, a characterization result for a set C to be smooth is given in terms of the τ C -approximation.

This note investigates the relationships between the generalized directional derivatives of the generalized distance function and the existence of the generalized nearest points in Banach spaces. It is proved that the generalized distance function associated with a closed bounded convex set having the Clark, Michel-Penot, Dini, or modified Dini derivative equals to 1 or −1 implies the existence of generalized nearest points. Also, new characterization theorems of (compact) locally uniformly sets are obtained.

Let (S,@S,@m) be a complete positive @s-finite measure space and let X be a Banach space. We are concerned with the proximinality problem for the best simultaneous approximations to two functions in L"p(S,@S,X). Let @S"0 be a sub-@s-algebra of @S and Y a nonempty locally weakly compact convex subset of X such that spanY@? and its dual have the Radon-Nikodym property. We prove that L"p(S,@S"0,Y) is N-simultaneous proximinal in L"p(S,@S,X) (with the additional assumption that (S,@S,@m) be finite for the case when p=1). Furthermore, for the special case when @S"0=@S, we show that the assumption that the dual of spanY@? has the Radon-Nikodym property can be removed.