v2007.09.13 - Convex Optimization

388 CHAPTER 6. EDM CONEa resemblance to EDM definition (705) whereS N h∆= { A ∈ S N | δ(A) = 0 } (56)is the symmetric hollow subspace (2.2.3) and whereS N⊥c = {u1 T + 1u T | u∈ R N } (1763)is the orthogonal complement of the geometric center subspace (E.7.2.0.2)S N c∆= {Y ∈ S N | Y 1 = 0} (1761)6.0.1 gravityEquality (1070) is equally important as the known isomorphisms (812) (813)(824) (825) relating the EDM cone EDM N to an N(N −1)/2-dimensionalface of S N + (5.6.1.1), or to S N−1+ (5.6.2.1). 6.1 Those isomorphisms havenever led to this equality (1070) relating the whole cones EDM N and S N + .Equality (1070) is not obvious from the various EDM matrix definitionssuch as (705) or (996) because inclusion must be proved algebraically in orderto establish equality; EDM N ⊇ S N h ∩ (S N⊥c − S N +). We will instead prove(1070) 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(724)is a discretized membership relation (2.13.4) between the EDM cone and itsordinary dual.6.1 Because both positive semidefinite cones are frequently in play, dimension is notated.