v2010.10.26 - Convex Optimization

498 CHAPTER 6. CONE OF DISTANCE MATRICESa resemblance to EDM definition (891) whereS N h = { A ∈ S N | δ(A) = 0 } (66)is the symmetric hollow subspace (2.2.3) and whereS N⊥c = {u1 T + 1u T | u∈ R N } (2000)is the orthogonal complement of the geometric center subspace (E.7.2.0.2)S N c = {Y ∈ S N | Y 1 = 0} (1998)6.0.1 gravityEquality (1268) is equally important as the known isomorphisms (1007)(1008) (1019) (1020) relating the EDM cone EDM N to positive semidefinitecone 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 relatingwhole cones EDM N and S N + .Equality (1268) is not obvious from the various EDM definitions such as(891) or (1191) because inclusion must be proved algebraically in order toestablish equality; EDM N ⊇ S N h ∩ (S N⊥c − S N +). We will instead prove (1268)using purely geometric methods.6.0.2 highlightIn6.8.1.7 we show: the Schoenberg criterion for discriminating Euclideandistance matricesD ∈ EDM N⇔{−VTN DV N ∈ S N−1+D ∈ S N h(910)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.