v2009.01.01 - Convex Optimization

130 CHAPTER 2. CONVEX GEOMETRY
Barker & Carlson [20,1] call the extreme directions a minimal generating
set for a pointed closed convex cone. A minimal set of generators is therefore
a conically independent set of generators, and vice versa, 2.43 for a pointed
closed convex cone.
An arbitrary collection of n or fewer distinct extreme directions from
pointed closed convex cone K ⊂ R n is not necessarily a linearly independent
set; e.g., dual extreme directions (431) from Example 2.13.11.0.3.
{≤ n extreme directions in R n } {l.i.}
Linear dependence of few extreme directions is another convex idea that
cannot be explained by a two-dimensional picture as Barvinok suggests
[25, p.vii]; indeed, it only first comes to light in four dimensions! But there
is a converse: [286,2.10.9]
{extreme directions} ⇐ {l.i. generators of closed convex K}
2.10.2.0.1 Example. Vertex-description of halfspace H about origin.
From n + 1 points in R n we can make a vertex-description of a convex
cone that is a halfspace H , where {x l ∈ R n , l=1... n} constitutes a
minimal set of generators for a hyperplane ∂H through the origin. An
example is illustrated in Figure 45. By demanding the augmented set
{x l ∈ R n , l=1... n + 1} be affinely independent (we want x n+1 not parallel
to ∂H), then
H = ⋃ (ζ x n+1 + ∂H)
ζ ≥0
= {ζ x n+1 + cone{x l ∈ R n , l=1... n} | ζ ≥0}
= cone{x l ∈ R n , l=1... n + 1}
(257)
a union of parallel hyperplanes. Cardinality is one step beyond dimension of
the ambient space, but {x l ∀l} is a minimal set of generators for this convex
cone H which has no extreme elements.
2.10.2.0.2 Exercise. Enumerating conically independent directions.
Describe a nonpointed polyhedral cone in three dimensions having more than
8 conically independent generators. (confer Table 2.10.0.0.1)
2.43 This converse does not hold for nonpointed closed convex cones as Table 2.10.0.0.1
implies; e.g., ponder four conically independent generators for a plane (case n=2).