v2009.01.01 - Convex Optimization

84 CHAPTER 2. CONVEX GEOMETRY
The point of intersection in C with a strictly supporting hyperplane
identifies an extreme point, but not vice versa. The nonempty intersection of
any supporting hyperplane with C identifies a face, in general, but not vice
versa. To acquire a converse, the concept exposed face requires introduction:
2.6.1 Exposure
2.6.1.0.1 Definition. Exposed face, exposed point, vertex, facet.
[173,A.2.3, A.2.4]
Fis an exposed face of an n-dimensional convex set C iff there is a
supporting hyperplane ∂H to C such that
F = C ∩ ∂H (150)
Only faces of dimension −1 through n −1 can be exposed by a
hyperplane.
An exposed point, the definition of vertex, is equivalent to a
zero-dimensional exposed face; the point of intersection with a strictly
supporting hyperplane.
Afacet is an (n −1)-dimensional exposed face of an n-dimensional
convex set C ; facets exist in one-to-one correspondence with the
(n −1)-dimensional faces. 2.26
{exposed points} = {extreme points}
{exposed faces} ⊆ {faces}
△
2.6.1.1 Density of exposed points
For any closed convex set C , its exposed points constitute a dense subset of
its extreme points; [266,18] [291] [286,3.6, p.115] dense in the sense [326]
that closure of that subset yields the set of extreme points.
For the convex set illustrated in Figure 27, point B cannot be exposed
because it relatively bounds both the facet AB and the closed quarter circle,
each bounding the set. Since B is not relatively interior to any line segment
in the set, then B is an extreme point by definition. Point B may be regarded
as the limit of some sequence of exposed points beginning at vertex C .
2.26 This coincidence occurs simply because the facet’s dimension is the same as the
dimension of the supporting hyperplane exposing it.