v2007.09.13 - Convex Optimization

5.14. FIFTH PROPERTY OF EUCLIDEAN METRIC 3775.14 Fifth property of Euclidean metricWe continue now with the question raised in5.3 regarding the necessityfor at least one requirement more than the four properties of the Euclideanmetric (5.2) to certify realizability of a bounded convex polyhedron or toreconstruct a generating list for it from incomplete distance information.There we saw that the four Euclidean metric properties are necessary forD ∈ EDM N in the case N = 3, but become insufficient when cardinality Nexceeds 3 (regardless of affine dimension).5.14.1 RecapitulateIn the particular case N = 3, −V T N DV N ≽ 0 (863) and D ∈ S 3 h are necessaryand sufficient conditions for D to be an EDM. By (865), triangle inequality isthen the only Euclidean condition bounding the necessarily nonnegative d ij ;and those bounds are tight. That means the first four properties of theEuclidean metric are necessary and sufficient conditions for D to be an EDMin the case N = 3 ; for i,j∈{1, 2, 3}√dij ≥ 0, i ≠ j√dij = 0, i = j√dij = √ d ji √dij≤ √ d ik + √ d kj , i≠j ≠k⇔ −V T N DV N ≽ 0D ∈ S 3 h⇔ D ∈ EDM 3(959)Yet those four properties become insufficient when N > 3.5.14.2 Derivation of the fifthCorrespondence between the triangle inequality and the EDM was developedin5.8.2 where a triangle inequality (865a) was revealed within theleading principal 2 ×2 submatrix of −VN TDV N when positive semidefinite.Our choice of the leading principal submatrix was arbitrary; actually,a unique triangle inequality like (768) corresponds to any one of the(N −1)!/(2!(N −1 − 2)!) principal 2 ×2 submatrices. 5.53 Assuming D ∈ S 4 hand −VN TDV N ∈ S 3 , then by the positive (semi)definite principal submatrices5.53 There are fewer principal 2×2 submatrices in −V T N DV N than there are triangles madeby four or more points because there are N!/(3!(N − 3)!) triangles made by point triples.The triangles corresponding to those submatrices all have vertex x 1 . (confer5.8.2.1)