v2009.01.01 - Convex Optimization

406 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX
5.9 Bridge: Convex polyhedra to EDMs
The criteria for the existence of an EDM include, by definition (798) (863),
the properties imposed upon its entries d ij by the Euclidean metric. From
5.8.1 and5.8.2, we know there is a relationship of matrix criteria to those
properties. Here is a snapshot of what we are sure: for i , j , k ∈{1... N}
(confer5.2)
√
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
δ(D) = 0
D T = D
(979)
all implied by D ∈ EDM N . In words, these four Euclidean metric properties
are necessary conditions for D to be a distance matrix. At the moment,
we have no converse. As of concern in5.3, we have yet to establish
metric requirements beyond the four Euclidean metric properties that would
allow D to be certified an EDM or might facilitate polyhedron or list
reconstruction from an incomplete EDM. We deal with this problem in5.14.
Our present goal is to establish ab initio the necessary and sufficient matrix
criteria that will subsume all the Euclidean metric properties and any further
requirements 5.41 for all N >1 (5.8.3); id est,
−V T N DV N ≽ 0
D ∈ S N h
}
⇔ D ∈ EDM N (817)
or for EDM definition (872),
}
Ω ≽ 0
√
δ(d) ≽ 0
⇔ D = D(Ω,d) ∈ EDM N (980)
5.41 In 1935, Schoenberg [270, (1)] first extolled matrix product −VN TDV N (958)
(predicated on symmetry and self-distance) specifically incorporating V N , albeit
algebraically. He showed: nonnegativity −y T VN TDV N y ≥ 0, for all y ∈ R N−1 , is necessary
and sufficient for D to be an EDM. Gower [136,3] remarks how surprising it is that such
a fundamental property of Euclidean geometry was obtained so late.