v2009.01.01 - Convex Optimization

5.14. FIFTH PROPERTY OF EUCLIDEAN METRIC 435
5.14 Fifth property of Euclidean metric
We continue now with the question raised in5.3 regarding the necessity
for at least one requirement more than the four properties of the Euclidean
metric (5.2) to certify realizability of a bounded convex polyhedron or to
reconstruct a generating list for it from incomplete distance information.
There we saw that the four Euclidean metric properties are necessary for
D ∈ EDM N in the case N = 3, but become insufficient when cardinality N
exceeds 3 (regardless of affine dimension).
5.14.1 Recapitulate
In the particular case N = 3, −V T N DV N ≽ 0 (964) and D ∈ S 3 h are necessary
and sufficient conditions for D to be an EDM. By (966), triangle inequality is
then the only Euclidean condition bounding the necessarily nonnegative d ij ;
and those bounds are tight. That means the first four properties of the
Euclidean metric are necessary and sufficient conditions for D to be an EDM
in 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 ≽ 0
D ∈ S 3 h
⇔ D ∈ EDM 3
(1061)
Yet those four properties become insufficient when N > 3.
5.14.2 Derivation of the fifth
Correspondence between the triangle inequality and the EDM was developed
in5.8.2 where a triangle inequality (966a) was revealed within the
leading 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 (861) corresponds to any one of the
(N −1)!/(2!(N −1 − 2)!) principal 2 ×2 submatrices. 5.54 Assuming D ∈ S 4 h
and −VN TDV N ∈ S 3 , then by the positive (semi)definite principal submatrices
5.54 There are fewer principal 2×2 submatrices in −V T N DV N than there are triangles made
by 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)