The problem of convexity of Chebyshev sets

Abstract

Contents Introduction §1. Definitions and notation §2. Reference theorems §3. Some results Chapter I. Characterization of Banach spaces by means of the relations between approximation properties of sets §1. Existence, uniqueness §2. Prom approximate compactness to 'sun'-property §3. From 'sun'-property to approximate compactness §4. Differentiability in the direction of the gradient is sufficient for Fréchet and Gâteaux differentiability §5. Sets with convex complement Chapter II. The structure of Chebyshev and related sets §1. The isolated point method §2. Restrictions of the type §3. The case where M is locally compact §4. The case where W lies in a hyperplane §5. Other cases Chapter III. Selected results §1. Some applications of the theory of monotone operators §2. A non-convex Chebyshev set in pre-Hilbert space §3. The example of Klee (discrete Chebyshev set) §4. A survey of some other results Conclusion Bibliography