On the convexity of the weakly compact Chebyshev sets in Banach spaces (Q1567336)

A subset \(M\) of a normed linear space \(X\) is called a Chebyshev set if to each point \(x\) of \(X\) there exists a unique point of \(M\) nearest to \(x\). The main result of this paper is a sufficient condition (almost smooth) for a Banach spaces \(X\) that every weakly compact Chebyshev set is convex. As an immediate consequence a former result of A. Brøndsted and A. L. Brown is partially extended and a known result in Hilbert spaces is proved in a different way. An important tool is to generalize the notion of extreme points to so-called \(n\)-extreme points. The paper includes all of the proofs.