Facial structure

The notions of face, extreme point and exposed point of a convex set were defined in Section 1.4. In the present section we shall study the boundary structure of closed convex sets in relation to these and similar or more specialized notions. We shall assume in the following that K ⊂ ℝn is a nonempty closed convex set.

An i-dimensional face of K is referred to as an i-face. By F(K) we denote the set of all faces and by Fi(K) the setofall i-faces of K. A face of dimension dim K – 1 is usually called a facet. The empty set ∅ and K itself are faces of K; the other faces are called proper. Conventionally, the empty face has dimension –1. It follows from the definition of a face and from Lemma 1.1.9 that the faces of K are closed. If F ≠ K is a face of K, then F ∩ relint K = ∅. (If z ∈ F ∩ relint K, we choose y ∈ K \ F. There is some x ∈ K with z ∈ relint [x,y]. Then [x,y] ⊂ F, a contradiction.) In particular, F ⊂ relbd K and dim F < dim K.