v2009.01.01 - Convex Optimization

446 CHAPTER 6. CONE OF DISTANCE MATRICES
a resemblance to EDM definition (798) where
S N h
∆
= { A ∈ S N | δ(A) = 0 } (59)
is the symmetric hollow subspace (2.2.3) and where
S N⊥
c = {u1 T + 1u T | u∈ R N } (1876)
is the orthogonal complement of the geometric center subspace (E.7.2.0.2)
S N c
∆
= {Y ∈ S N | Y 1 = 0} (1874)
6.0.1 gravity
Equality (1173) is equally important as the known isomorphisms (912) (913)
(924) (925) relating the EDM cone EDM N to positive semidefinite cone S N−1
+
(5.6.2.1) or to an N(N −1)/2-dimensional face of S N + (5.6.1.1). 6.1 But
those isomorphisms have never led to this equality relating whole cones
EDM N and S N + .
Equality (1173) is not obvious from the various EDM definitions such as
(798) or (1099) because inclusion must be proved algebraically in order to
establish equality; EDM N ⊇ S N h ∩ (S N⊥
c − S N +). We will instead prove (1173)
using purely geometric methods.
6.0.2 highlight
In6.8.1.7 we show: the Schoenberg criterion for discriminating Euclidean
distance matrices
D ∈ EDM N
⇔
{
−V
T
N DV N ∈ S N−1
+
D ∈ S N h
(817)
is a discretized membership relation (2.13.4, dual generalized inequalities)
between the EDM cone and its ordinary dual.
6.1 Because both positive semidefinite cones are frequently in play, dimension is explicit.