v2009.01.01 - Convex Optimization

6.2. POLYHEDRAL BOUNDS 449
This cone is more easily visualized in the isomorphic vector subspace
R 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 ; a
halfline in a subspace of the realization in Figure 122.
The EDM cone in the case N = 3 is a circular cone in R 3 illustrated in
Figure 114(a)(d); rather, the set of all matrices
⎡
D = ⎣
0 d 12 d 13
d 12 0 d 23
d 13 d 23 0
⎤
⎦ ∈ EDM 3 (1085)
makes a circular cone in this dimension. In this case, the first four Euclidean
metric properties are necessary and sufficient tests to certify realizability
of triangles; (1061). Thus triangle inequality property 4 describes three
halfspaces (967) whose intersection makes a polyhedral cone in R 3 of
realizable √ d ij (absolute distance); an isomorphic subspace representation
of the set of all EDMs D in the natural coordinates
⎡ √ √ ⎤
0 d12 d13
◦√
D =
∆ √d12 √
⎣ 0 d23 ⎦
√d13 √
(1086)
d23 0
illustrated in Figure 114(b).
6.2 Polyhedral bounds
The convex cone of EDMs is nonpolyhedral in d ij for N > 2 ; e.g.,
Figure 114(a). Still we found necessary and sufficient bounding polyhedral
relations consistent with EDM cones for cardinality N = 1, 2, 3, 4:
N = 3. Transforming distance-square coordinates d ij by taking their positive
square root provides polyhedral cone in Figure 114(b); polyhedral
because an intersection of three halfspaces in natural coordinates
√
dij is provided by triangle inequalities (967). This polyhedral
cone implicitly encompasses necessary and sufficient metric properties:
nonnegativity, self-distance, symmetry, and triangle inequality.