Separating and Supporting Hyperplanes

Abstract

Now that we have a solid background in Euclidean geometry, we can go deeper into our study of convex sets begun in Chapter 3. This chapter is devoted to a thorough study of separating and supporting hyperplanes. We prove two geometric versions of the Hahn–Banach theorem, from which we derive separation results for various kinds of pairs of convex sets (open, closed, compact). We prove various versions of Farkas’s lemma, a basic result in the theory of linear programming. We also discuss supporting hyperplanes and prove an important proposition due to Minkowski.