Convex Sets

Abstract

In this chapter we introduce the fundamental notion of the convexity of a set and establish various properties of convex sets. The key result is Theorem 3.14, which asserts that every nonempty closed convex subset C of $\mathcal{H}$ is a Chebyshev set, i.e., that every point in $\mathcal{H}$ possesses a unique best approximation from C, and which provides a characterization of this best approximation.