2 - Convex Sets

Summary

This chapter introduces the fundamental notion of a convex set. It establishes basic structural properties of convex sets, illustrated via examples throughout. The chapter gives equal emphasis on the analytic as well as discrete or combinatorial aspects of convexity. Topics include foundational results like the Separating and Supporting Hyperplane theorems, polarity, the combinatorial theorems of Caratheodory, Radon and Helly, and the basic theory of polyhedra and ellipsoids.