v2010.10.26 - Convex Optimization

5.2. FIRST METRIC PROPERTIES 397to correspond to D in (880). But such a list is not unique because anyrotation, reflection, or translation (5.5) of those points would produce thesame EDM D .5.2 First metric propertiesFor i,j =1... N , absolute distance between points x i and x j must satisfythe defining requirements imposed upon any metric space: [227,1.1][258,1.7] namely, for Euclidean metric √ d ij (5.4) in R n1. √ d ij ≥ 0, i ≠ j nonnegativity2. √ d ij = 0 ⇔ x i = x j self-distance3. √ d ij = √ d ji symmetry4. √ d ij ≤ √ d ik + √ d kj , i≠j ≠k triangle inequalityThen all entries of an EDM must be in concord with these Euclidean metricproperties: specifically, each entry must be nonnegative, 5.2 the main diagonalmust be 0 , 5.3 and an EDM must be symmetric. The fourth propertyprovides upper and lower bounds for each entry. Property 4 is true moregenerally when there are no restrictions on indices i,j,k , but furnishes nonew information.5.3 ∃ fifth Euclidean metric propertyThe four properties of the Euclidean metric provide information insufficientto certify that a bounded convex polyhedron more complicated than atriangle has a Euclidean realization. [162,2] Yet any list of points or thevertices 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. [258,1.6] Kreyszig callsthese first metric properties axioms of the metric; [227, p.4] Blumenthal refers to them aspostulates. [50, p.15]