v2009.01.01 - Convex Optimization

5.2. FIRST METRIC PROPERTIES 347
correspond to D in (787). Such a list is not unique because any rotation,
reflection, or translation (5.5) of the points in Figure 92 would produce the
same EDM D .
5.2 First metric properties
For i,j =1... N , distance between points x i and x j must satisfy the
defining requirements imposed upon any metric space: [197,1.1] [222,1.7]
namely, for Euclidean metric √ d ij (5.4) in R n
1. √ d ij ≥ 0, i ≠ j nonnegativity
2. √ d ij = 0, i = j self-distance
3. √ d ij = √ d ji symmetry
4. √ d ij ≤ √ d ik + √ d kj , i≠j ≠k triangle inequality
Then all entries of an EDM must be in concord with these Euclidean metric
properties: specifically, each entry must be nonnegative, 5.2 the main diagonal
must be 0 , 5.3 and an EDM must be symmetric. The fourth property
provides upper and lower bounds for each entry. Property 4 is true more
generally when there are no restrictions on indices i,j,k , but furnishes no
new information.
5.3 ∃ fifth Euclidean metric property
The four properties of the Euclidean metric provide information insufficient
to certify that a bounded convex polyhedron more complicated than a
triangle has a Euclidean realization. [136,2] Yet any list of points or the
vertices of any bounded convex polyhedron must conform to the properties.
5.2 Implicit from the terminology, √ d ij ≥ 0 ⇔ d ij ≥ 0 is always assumed.
5.3 What we call self-distance, Marsden calls nondegeneracy. [222,1.6] Kreyszig calls
these first metric properties axioms of the metric; [197, p.4] Blumenthal refers to them as
postulates. [44, p.15]