Farthest points in W*-compact sets

Abstract

We show that while farthest points always exist in w* -compact sets in duals to RadonNikodym spaces, this is generally not the case in dual Radon-Nikodym spaces. We also show how to characterise weak compactness in terms of farthest points.The purpose of this note is to find under what conditions the Edelstein-AsplundLau results on the existence of farthest points in weakly compact sets can be extended to w* -compact sets.To fix our notation, let C be a norm closed bounded subset of a real Banach space X and x be an element of X. We defineand call r(x) the farthest distance from x to C. Equivalently, r(x) is the radius of the smallest ball of centre x , containing C. The function r is convex as supremum of such functions, and continuous since \r(x) -r(y)\ ^ ||x -y\\, for all x , y £ X . A point z £ C is called a farthest point of C if there exists x £ X such that ||x -z|| = r(x). The existence of a farthest point of C is equivalent to the fact that the set is non-empty. Since it follows that any farthest point of a convex set C in a locally uniformly rotund space is a strongly exposed point of C, the notion of a farthest point is widely used in the study of the extreme structure of sets. In fact, it was the first discovered method of obtaining exposed points in sets [10]. We refer the reader to [3] and [5] for unexplained notions.Extending the results of [6] and [1], Lau showed [9] that if C is a weakly compact set in a Banach space X, then the set D defined above is dense in X and thus, in particular, C has farthest points. Other extensions of the results of [6] and [1] may be found in [11]. The connection between farthest points and strongly exposed points in

Some remarks on farthest points

“…The following result extends Lemma 2 in [7] (there, the case in which S := B X was covered), and has again a simple proof based on Lemma 10. Together with Proposition 19, they contain also Theorem 2.6 (the part of the statement about farthest points) in [13].…”

“…The following result contains Proposition 3 in [7] and extends Proposition 1 in [22]. For a reference on RNP spaces see, e.g., [11,Ch.…”

“…For previous results in this area, see, in particular, [1,8,9,13,15,19,21,22], and [7], among others, and for a survey see, e.g., [4].…”

“…The following result is the LUR counterpart of a result about farthest points and strict convexity in [7,Proposition 5].…”

Farthest points and the farthest distance map

“…(b) li X has the Radon-Nikodym Property, by [4,Proposition 3], each u/*-compact set K C X* is densely remotal. Thus, necessity can be proved as in (a).…”