Chapter 8 - Convex sets and convex cones

Abstract

This chapter introduces the concept of convexity and convex cones and discusses their properties. The concept of convexity that is now introduced is of great importance in the study of optimization problems. Before defining convex sets and their properties, the chapter first gives some definitions of line, line segment, half line, hyperplane, and half-spaces. A set S in Rn is said to be convex if the line segment joining any two points in the set also belongs to the set. Linear subspaces, triangles, and spheres are some simple examples of convex sets. In particular the empty set ϕ, sets with a single point only and Rn are convex. Extending the concept of convex combination, the chapter generalizes its definition. A vector X in Rn is said to be a convex combination of the vectors X1, X2,.. Xm ∊ Rn if there exist real numbers λ1, λ 2,… λm.