v2007.09.13 - Convex Optimization

5.9. BRIDGE: CONVEX POLYHEDRA TO EDMS 351The elliptope for dimension n = 2 is a line segment in isometricallyisomorphic R n(n+1)/2 (Figure 87). Obviously, cone(E n )≠ S n + . The elliptopefor dimension n = 3 is realized in Figure 86.5.9.1.0.2 Lemma. Hypersphere. [15,4] (confer bullet p.302)Matrix A = [A ij ]∈ S N belongs to the elliptope in S N iff there exist N pointsp on the boundary of a hypersphere having radius 1 in R rank A such that‖p i − p j ‖ = √ 2 √ 1 − A ij , i,j=1... N (882)There is a similar theorem for Euclidean distance matrices:We derive matrix criteria for D to be an EDM, validating (724) usingsimple geometry; distance to the polyhedron formed by the convex hull of alist of points (65) in Euclidean space R n .⋄5.9.1.0.3 EDM assertion.D is a Euclidean distance matrix if and only if D ∈ S N h and distances-squarefrom the origin{‖p(y)‖ 2 = −y T V T NDV N y | y ∈ S − β} (883)correspond to points p in some bounded convex polyhedronP − α = {p(y) | y ∈ S − β} (884)having N or fewer vertices embedded in an r-dimensional subspace A − αof R n , where α ∈ A = aff P and where the domain of linear surjection p(y)is the unit simplex S ⊂ R N−1+ shifted such that its vertex at the origin istranslated to −β in R N−1 . When β = 0, then α = x 1 .⋄In terms of V N , the unit simplex (253) in R N−1 has an equivalentrepresentation:S = {s ∈ R N−1 | √ 2V N s ≽ −e 1 } (885)where e 1 is as in (721). Incidental to the EDM assertion, shifting theunit-simplex domain in R N−1 translates the polyhedron P in R n . Indeed,