v2009.01.01 - Convex Optimization

86 CHAPTER 2. CONVEX GEOMETRY
2.6.1.2 Face transitivity and algebra
Faces 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 a
face of F 3 . (The parallel statement for exposed faces is false. [266,18])
For example, any extreme point of F 2 is an extreme point of F 3 ; in
this example, F 2 could be a face exposed by a hyperplane supporting
polyhedron F 3 . [194, def.115/6, p.358] Yet it is erroneous to presume that
a face, of dimension 1 or more, consists entirely of extreme points, nor is a
face of dimension 2 or more entirely composed of edges, and so on.
For the polyhedron in R 3 from Figure 16, for example, the nonempty
faces exposed by a hyperplane are the vertices, edges, and facets; there
are no more. The zero-, one-, and two-dimensional faces are in one-to-one
correspondence with the exposed faces in that example.
Define the smallest face F that contains some element G of a convex
set C :
F(C ∋G) (151)
videlicet, C ⊇ F(C ∋G) ∋ G . An affine set has no faces except itself and the
empty set. The smallest face that contains G of the intersection of convex
set C with an affine set A [206,2.4] [207]
F((C ∩ A)∋G) = F(C ∋G) ∩ A (152)
equals the intersection of A with the smallest face that contains G of set C .
2.6.1.3 Boundary
The 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
[173,A.2.1.2]
rel∂C = C \ rel int C (153)
the boundary relative to the affine hull of C , conventionally equivalent to: