v2009.01.01 - Convex Optimization

190 CHAPTER 2. CONVEX GEOMETRY
A natural question pertains to whether a theory of unique coordinates,
like biorthogonal expansion, is extensible to proper cones whose extreme
directions number in excess of ambient spatial dimensionality.
2.13.11.0.4 Theorem. Conic coordinates.
With respect to vector v in some finite-dimensional Euclidean space R n ,
define a coordinate t ⋆ v of point x in full-dimensional pointed closed convex
cone K
t ⋆ v(x) ∆ = sup{t∈ R | x − tv ∈ K} (432)
Given points x and y in cone K , if t ⋆ v(x)= t ⋆ v(y) for each and every extreme
direction v of K then x = y .
⋄
2.13.11.0.5 Proof. Vector x −t ⋆ v must belong to the cone
boundary ∂K by definition (432). So there must exist a nonzero vector λ that
is inward-normal to a hyperplane supporting cone K and containing x −t ⋆ v ;
id est, by boundary membership relation for full-dimensional pointed closed
convex cones (2.13.2)
x−t ⋆ v ∈ ∂K ⇔ ∃ λ ≠ 0 〈λ, x−t ⋆ v〉 = 0, λ ∈ K ∗ , x−t ⋆ v ∈ K (298)
where
K ∗ = {w ∈ R n | 〈v , w〉 ≥ 0 for all v ∈ G(K)} (332)
is the full-dimensional pointed closed convex dual cone. The set G(K) , of
possibly infinite cardinality N , comprises generators for cone K ; e.g., its
extreme directions which constitute a minimal generating set. If x −t ⋆ v
is nonzero, any such vector λ must belong to the dual cone boundary by
conjugate boundary membership relation
λ ∈ ∂K ∗ ⇔ ∃ x−t ⋆ v ≠ 0 〈λ, x−t ⋆ v〉 = 0, x−t ⋆ v ∈ K , λ ∈ K ∗ (299)
where
K = {z ∈ R n | 〈λ, z〉 ≥ 0 for all λ ∈ G(K ∗ )} (331)
This description of K means: cone K is an intersection of halfspaces whose
inward-normals are generators of the dual cone. Each and every face of
cone K (except the cone itself) belongs to a hyperplane supporting K . Each
and every vector x −t ⋆ v on the cone boundary must therefore be orthogonal
to an extreme direction constituting generators G(K ∗ ) of the dual cone.