Mathematical techniques

Abstract

Let X be a (real) vector space (e.g. ∝ⁿ or C(I)). A linear combination λ₁ + λ₂ x ₂ + … + λx _r x _r of the vectors x ₁, x ₂, …, x _r in X is called a convex combination of these vectors if the real numbers λ₁ satisfy (Ai)λ _i ≥ 0, and Σr/i≡ 1. The convex hull co E of a set E ⊂ X is the set of all convex combinations of finite sets of points in E. In particular, co x, yis the straight-line segment [x, y] joining the points x and y. A set E ⊂ X is convex if [x, y] ⊂ E whenever x,y ⊂E. (Equivalently, by an induction on the number of points, E is convex iff E = co E.)