Extreme Points and Directions for Convex Sets

Abstract

Let S be a convex set in R n . Two vectors (points) x1,x2 ∈S are said to be distinct if x1 ≠ αx2 for any 0 < α ∈ R (i.e, one vector cannot be written as a positive multiple of the other). In this regard, a vector x ∈ S is an extreme point of S if it is not the mean of two other distinct vectors in S, e.g., if extreme x is the mean of x1, x 2 ∈ S, then x = x l = x 2 . Equivalently, a point x ∈ S is extreme if x cannot be expressed as a positive convex combination of two distinct points in S. Thus x is an extreme point of S if and only if x =λx 1 + (1-λ)x2, 0 < λ < 1, and x 1 , x2 ∈ S implies x = xl = x2. Hence there is no way to express x as a positive convex combination of x1, x2 except by taking x= xl = x2. Clearly an extreme point of a convex set S cannot lie between and two other points of the set.