v2009.01.01 - Convex Optimization

132 CHAPTER 2. CONVEX GEOMETRY
2.11 When extreme means exposed
For any convex polyhedral set in R n having nonempty interior, distinction
between the terms extreme and exposed vanishes [286,2.4] [96,2.2] for
faces of all dimensions except n ; their meanings become equivalent as we
saw in Figure 16 (discussed in2.6.1.2). In other words, each and every face
of any polyhedral set (except the set itself) can be exposed by a hyperplane,
and vice versa; e.g., Figure 20.
Lewis [212,6] [186,2.3.4] claims nonempty extreme proper subsets and
the exposed subsets coincide for S n + ; id est, each and every face of the positive
semidefinite cone, whose dimension is less than the dimension of the cone,
is exposed. A more general discussion of cones having this property can be
found in [296]; e.g., the Lorentz cone (160) [19,II.A].
2.12 Convex polyhedra
Every polyhedron, such as the convex hull (78) of a bounded list X , can
be expressed as the solution set of a finite system of linear equalities and
inequalities, and vice versa. [96,2.2]
2.12.0.0.1 Definition. Convex polyhedra, halfspace-description.
A convex polyhedron is the intersection of a finite number of halfspaces and
hyperplanes;
P = {y | Ay ≽ b, Cy = d} ⊆ R n (258)
where coefficients A and C generally denote matrices. Each row of C is a
vector normal to a hyperplane, while each row of A is a vector inward-normal
to a hyperplane partially bounding a halfspace.
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By the halfspaces theorem in2.4.1.1.1, a polyhedron thus described is a
closed convex set having possibly empty interior; e.g., Figure 16. Convex
polyhedra 2.44 are finite-dimensional comprising all affine sets (2.3.1),
polyhedral cones, line segments, rays, halfspaces, convex polygons, solids
[194, def.104/6, p.343], polychora, polytopes, 2.45 etcetera.
2.44 We consider only convex polyhedra throughout, but acknowledge the existence of
concave polyhedra. [326, Kepler-Poinsot Solid]
2.45 Some authors distinguish bounded polyhedra via the designation polytope. [96,2.2]