v2009.01.01 - Convex Optimization

348 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX
5.3.0.0.1 Example. Triangle.
Consider the EDM in (787), but missing one of its entries:
⎡
0
⎤
1 d 13
D = ⎣ 1 0 4 ⎦ (788)
d 31 4 0
Can we determine unknown entries of D by applying the metric properties?
Property 1 demands √ d 13 , √ d 31 ≥ 0, property 2 requires the main diagonal
be 0, while property 3 makes √ d 31 = √ d 13 . The fourth property tells us
1 ≤ √ d 13 ≤ 3 (789)
Indeed, described over that closed interval [1, 3] is a family of triangular
polyhedra whose angle at vertex x 2 varies from 0 to π radians. So, yes we
can determine the unknown entries of D , but they are not unique; nor should
they be from the information given for this example.
5.3.0.0.2 Example. Small completion problem, I.
Now consider the polyhedron in Figure 93(b) formed from an unknown
list {x 1 ,x 2 ,x 3 ,x 4 }. The corresponding EDM less one critical piece of
information, d 14 , is given by
⎡
⎤
0 1 5 d 14
D = ⎢ 1 0 4 1
⎥
⎣ 5 4 0 1 ⎦ (790)
d 14 1 1 0
From metric property 4 we may write a few inequalities for the two triangles
common to d 14 ; we find
√
5−1 ≤
√
d14 ≤ 2 (791)
We cannot further narrow those bounds on √ d 14 using only the four metric
properties (5.8.3.1.1). Yet there is only one possible choice for √ d 14 because
points x 2 ,x 3 ,x 4 must be collinear. All other values of √ d 14 in the interval
[ √ 5−1, 2] specify impossible distances in any dimension; id est, in this
particular example the triangle inequality does not yield an interval for √ d 14
over which a family of convex polyhedra can be reconstructed.