1 - Convex Sets and Functions

Abstract

Convex sets and convex functions are studied in this chapter in the setting of n-dimensional Euclidean space Rn. Convexity is an attractive subject to study, for many reasons; it draws upon geometry, analysis, linear algebra, and topology, and it has a role to play in such topics as classical optimal control theory, game theory, linear programming, and convex programming. If one is familiar with functional analysis, he or she will be able to generalize the main results to the case of infinite dimensional functional spaces. These results play a decisive role in obtaining the main results. The basic idea in convexity is that a convex function on Rn can be identified with a convex subset of Rn+1, which is called the epigraph of the given function. This identification makes it easy to move back and forth between geometrical and analytical approaches.