v2007.11.26 - Convex Optimization

404 CHAPTER 6. CONE OF DISTANCE MATRICESa resemblance to EDM definition (734) whereS N h∆= { A ∈ S N | δ(A) = 0 } (57)is the symmetric hollow subspace (2.2.3) and whereS N⊥c = {u1 T + 1u T | u∈ R N } (1794)is the orthogonal complement of the geometric center subspace (E.7.2.0.2)S N c∆= {Y ∈ S N | Y 1 = 0} (1792)6.0.1 gravityEquality (1100) is equally important as the known isomorphisms (841) (842)(853) (854) 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 (1100) relating the whole cones EDM N and S N + .Equality (1100) is not obvious from the various EDM matrix definitionssuch as (734) or (1026) because inclusion must be proved algebraically inorder to establish equality; EDM N ⊇ S N h ∩ (S N⊥c − S N +). We will insteadprove (1100) 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(753)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.