v2010.10.26 - Convex Optimization

5.14. FIFTH PROPERTY OF EUCLIDEAN METRIC 487reconstruct 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 (1059) and D ∈ S 3 h are necessaryand sufficient conditions for D to be an EDM. By (1061), 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(1156)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 (1061a) 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 (956) corresponds to any one of the(N −1)!/(2!(N −1 − 2)!) principal 2 ×2 submatrices. 5.59 Assuming D ∈ S 4 hand −VN TDV N ∈ S 3 , then by the positive (semi)definite principal submatricestheorem (A.3.1.0.4) it is sufficient to prove: all d ij are nonnegative, alltriangle inequalities are satisfied, and det(−VN TDV N) is nonnegative. WhenN = 4, in other words, that nonnegative determinant becomes the fifth andlast Euclidean metric requirement for D ∈ EDM N . We now endeavor toascribe geometric meaning to it.5.59 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)