v2007.11.26 - Convex Optimization

80 CHAPTER 2. CONVEX GEOMETRY2.6.1.2 Face transitivity and algebraFaces of a convex set enjoy transitive relation. If F 1 is a face (an extreme set)of F 2 which in turn is a face of F 3 , then it is always true that F 1 is aface of F 3 . (The parallel statement for exposed faces is false. [232,18])For example, any extreme point of F 2 is an extreme point of F 3 ; inthis example, F 2 could be a face exposed by a hyperplane supportingpolyhedron F 3 . [164, def.115/6, p.358] Yet it is erroneous to presume thata face, of dimension 1 or more, consists entirely of extreme points, nor is aface of dimension 2 or more entirely composed of edges, and so on.For the polyhedron in R 3 from Figure 14, for example, the nonemptyfaces exposed by a hyperplane are the vertices, edges, and facets; thereare no more. The zero-, one-, and two-dimensional faces are in one-to-onecorrespondence with the exposed faces in that example.Define the smallest face F that contains some element G of a convexset C :F(C ∋G) (139)videlicet, C ⊇ F(C ∋G) ∋ G . An affine set has no faces except itself and theempty set. The smallest face that contains G of the intersection of convexset C with an affine set A [176,2.4]F((C ∩ A)∋G) = F(C ∋G) ∩ A (140)equals the intersection of A with the smallest face that contains G of set C .2.6.1.3 BoundaryThe classical definition of boundary of a set C presumes nonempty interior:∂ C = C \ int C (14)More suitable for the study of convex sets is the relative boundary; defined[148,A.2.1.2]rel∂C = C \ rel int C (141)the boundary relative to the affine hull of C , conventionally equivalent to: