Principles of Mathematics in Operations Research

7

Convex Sets

This chapter is compiled to present a brief summary of the most important

concepts related to convex sets. Following the basic definitions, we will con-

centrate on supporting and separating hyperplanes, extreme points and poly-

topes.

7.1 Preliminaries

Definition 7.1.1 A set X in K" is said to be convex if

Mxi,X2 G X and Va e R+,0 < a < 1, the point ax\ + (1 - a)a;2 € X.

CONVEX NON-CONVEX

Fig. 7.1. Convexity

Remark 7.1.2 Geometrically speaking, X is convex if for any points X\,X2 £
X, the line segment joining these two points is also in the set. This is illus-
trated in Figure 7.1.

Definition 7.1.3 A point x € X is an extreme point of the convex set X if
and only if