Approximative compactness and continuity of metric projections

Abstract

In the paper "Some remarks on approximative compactness", Rev.Roumaine Math. Pyres Appl. 9 (196U), Ivan Singer proved that if K is an approximatively compact Chebyshev set in a metric space, then the metric projection onto K is continuous. The object of this paper is to show that though, in general, the continuity of the metric projection supported by a Chebyshev set does not imply that the set is approximatively compact, it is indeed so in a large class of Banach spaces, including the locally uniformly convex spaces. It is also proved that in such a space X the metric projection onto a Chebyshev set is continuous on a set dense in X .

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

“…The main results obtained in this section, which themselves have independent interest, are improvements and extensions of some known ones due to [33,30,19,24]. We next present in the last section several equivalent conditions (such as D-approximate compactness of the set C, continuity or maximal monotonicity of the Bregman projection Π g C , and the differentiability of the Bregman distance function D g C , etc.)…”

“…This result was extended in [34] by Vlasov to the setting of Banach spaces with round dual spaces. For the details and other related results, the readers are referred to [19,22,33,37,30] and the surveys [5,35]. Note that the continuity of the projection P C plays a key role in the study mentioned above.…”

Some Results on Fixed and Best Proximity Points of Multivalued Cyclic Self-Mappings with a Partial Order

“…is nonempty, bounded and closed; p j   and the given p i  . Thus, one gets from (3.1), since X A j  is nonempty, bounded and closed, and then boundedly compact, and also approximatively compact with respect to 1  j A ( [5], [27]), that: . The uniqueness property of each of those p best proximity points j j x T x  in each of the subsets X A j  follows from their uniqueness as fixed points of the restricted self-mappings…”

Strong Proximinality in Metric Spaces

“…It is well-known (see [5] [13]) Let X be the dual space of the normed linear space constructed by Klee [10] by suitable renorming of l 2 . Lambert (unpublished) (see [13]) has shown that in the space X the metric projection P W supported by any Chebyshev subspace W of X is continuous. However, X does not satisfy the Effimov-Steckin property and hence contains a closed hyperplane K which is not approximatively compact (see [14], Theorem 3).…”