v2007.09.17 - Convex Optimization

382 CHAPTER 5. EUCLIDEAN DISTANCE MATRIXBecause point labelling is arbitrary, this fifth Euclidean metricrequirement must apply to each of the N points as though each were in turnlabelled x 1 ; hence the new index k in (969). Just as the triangle inequalityis the ultimate test for realizability of only three points, the relative-angleinequality is the ultimate test for only four. For four distinct points, thetriangle inequality remains a necessary although penultimate test; (5.4.3)Four Euclidean metric properties (5.2).Angle θ inequality (703) or (969).⇔ −V T N DV N ≽ 0D ∈ S 4 h⇔ D = D(Θ) ∈ EDM 4The relative-angle inequality, for this case, is illustrated in Figure 93.5.14.2.2 Beyond the fifth metric property(970)When cardinality N exceeds 4, the first four properties of the Euclideanmetric and the relative-angle inequality together become insufficientconditions for realizability. In other words, the four Euclidean metricproperties and relative-angle inequality remain necessary but become asufficient test only for positive semidefiniteness of all the principal 3 × 3submatrices [sic] in −VN TDV N . Relative-angle inequality can be consideredthe ultimate test only for realizability at each vertex x k of each and everypurported tetrahedron constituting a hyperdimensional body.When N = 5 in particular, relative-angle inequality becomes thepenultimate Euclidean metric requirement while nonnegativity of thenunwieldy det(Θ T Θ) corresponds (by the positive (semi)definite principalsubmatrices theorem inA.3.1.0.4) to the sixth and last Euclidean metricrequirement. Together these six tests become necessary and sufficient, andso on.Yet for all values of N , only assuming nonnegative d ij , relative-anglematrix inequality in (883) is necessary and sufficient to certify realizability;(5.4.3.1)Euclidean metric property 1 (5.2).Angle matrix inequality Ω ≽ 0 (779).⇔ −V T N DV N ≽ 0D ∈ S N h⇔ D = D(Ω,d) ∈ EDM N(971)Like matrix criteria (704), (728), and (883), the relative-angle matrixinequality and nonnegativity property subsume all the Euclidean metricproperties and further requirements.