CHAPTER 1.1 - Characterizations of Convex Sets

Abstract

This chapter discusses the characterizations of convex sets. In Euclidean spaces convex sets have a supporting hyperplane at each boundary point; they are locally convex; and they possess a nearest point map. Furthermore, for convex sets and convex functions in Euclidian spaces four more aspects might be considered, such as, contact with affine subspaces, contact with some other classes of sets, metric properties, and other criteria. The Chebyshev property of convex sets doesn't allow easy generalizations, not even to finite dimensional Banach spaces. Klee constructs strange non-convex Chebyshev sets in perfectly harmless looking Banach spaces, under appropriate assumptions about the underlying set theory. An important class of sets has emerged in an attempt to expand the horizon of classical convexity—the sets of positive reach, which enjoy a local Chebyshev property. When characterizing properties of convex sets in analysis and differential geometry five topics need to be covered—convex functions, the Weyl problem, the Minkowski problem, convexity on the Grassmann cones, and quasiconvex functions. A convex surface is the boundary of a convex compact set with nonempty interior, and the local convexity of ƒ can be expressed by means of the Hessian, or by using sectional curvatures.