Convex sets, separation theorems, and nonconvex sets in R N

Abstract

Definition A set of points S in R N is said to be convex if the line segment between any two points of the set is completely included in the set, that is, S is convex if x, y ∈ S implies { z | z = α x + (1 - α) y , 0 ≤ α ≤ 1} ⊆ S . S is said to be strictly convex if x, y ∈ S, x ≠ y , 0 implies α x + (1 - α) y ∈ interior S . The notion of convexity is that a set is convex if it is connected, has no holes on the inside, and has no indentations on the boundary. Figure 8.1 displays convex and nonconvex sets. A set is strictly convex if it is convex and has a continuous strict curvature (no flat segments) on the boundary. Properties of convex sets Let C 1 and C 2 be convex subsets of R N . Then C 1 ∩ C 2 is convex , C 1 + C 2 is convex , C 1 is convex . Proof See Exercise 8.1. QED The concept of convexity of a set in R N is essential in mathematical economic analysis. This reflects the importance of continuous point-valued optimizing behavior. To understand the importance of convexity, consider for a moment what will happen when it is absent. Suppose widgets are consumed only in discrete lots of 100. The insistence on discrete lots is a nonconvexity. Suppose a typical widget eater at some prices to be indifferent between buying a lot of 100 and buying 0.