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  • Print publication year: 2011
  • Online publication date: June 2012

8 - Convex sets, separation theorems, and nonconvex sets in RN

from B - Mathematics

Summary

DefinitionA set of points S in RN 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, yS implies {z | z = αx + (1 - α)y, 0 ≤ α ≤ 1} ⊆ S.

S is said to be strictly convex if x, yS, xy, 0 < α ≤ 1 implies αx + (1 - α)yinterior 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 setsLet C1and C2be convex subsets of RN. Then

C1C2is convex,

C1 + C2is convex,

C1is convex.

Proof See Exercise 8.1. QED

The concept of convexity of a set in RN 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.