v2007.11.26 - Convex Optimization

6.2. POLYHEDRAL BOUNDS 407This cone is more easily visualized in the isomorphic vector subspaceR N(N−1)/2 corresponding to S N h :In the case N = 1 point, the EDM cone is the origin in R 0 .In the case N = 2, the EDM cone is the nonnegative real line in R ; ahalfline in a subspace of the realization in Figure 108.The EDM cone in the case N = 3 is a circular cone in R 3 illustrated inFigure 100(a)(d); rather, the set of all matrices⎡D = ⎣0 d 12 d 13d 12 0 d 23d 13 d 23 0⎤⎦ ∈ EDM 3 (1013)makes a circular cone in this dimension. In this case, the first four Euclideanmetric properties are necessary and sufficient tests to certify realizabilityof triangles; (989). Thus triangle inequality property 4 describes threehalfspaces (896) whose intersection makes a polyhedral cone in R 3 ofrealizable √ d ij (absolute distance); an isomorphic subspace representationof the set of all EDMs D in the natural coordinates⎡ √ √ ⎤0 d12 d13◦√D =∆ √d12 √⎣ 0 d23 ⎦√d13 √(1014)d23 0illustrated in Figure 100(b).6.2 Polyhedral boundsThe convex cone of EDMs is nonpolyhedral in d ij for N > 2 ; e.g.,Figure 100(a). Still we found necessary and sufficient bounding polyhedralrelations consistent with EDM cones for cardinality N = 1, 2, 3, 4:N = 3. Transforming distance-square coordinates d ij by taking their positivesquare root provides polyhedral cone in Figure 100(b); polyhedralbecause an intersection of three halfspaces in natural coordinates√dij is provided by triangle inequalities (896). This polyhedralcone implicitly encompasses necessary and sufficient metric properties:nonnegativity, self-distance, symmetry, and triangle inequality.