v2006.03.09 - Convex Optimization

4.2. FIRST METRIC PROPERTIES 217correspond to D in (449). Such a list is not unique because any rotation,reflection, or translation (4.5) of the points in Figure 56 would produce thesame EDM D .4.2 First metric propertiesFor i,j = 1... N , the Euclidean distance between points x i and x j mustsatisfy the requirements imposed by any metric space: [140,1.1] [160,1.7]1. √ d ij ≥ 0, i ≠ j nonnegativity2. √ d ij = 0, i = j self-distance3. √ d ij = √ d ji symmetry4. √ d ij ≤ √ d ik + √ d kj , i≠j ≠k triangle inequalitywhere √ d ij is the Euclidean metric in R n (4.4). Then all entries of an EDMmust be in concord with these Euclidean metric properties: specifically, eachentry must be nonnegative, 4.2 the main diagonal must be 0 , 4.3 and an EDMmust be symmetric. The fourth property provides upper and lower bounds foreach entry. Property 4 is true more generally when there are no restrictionson indices i,j,k , but furnishes no new information.4.3 ∃ 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. [90,2] Yet any list of points or thevertices of any bounded convex polyhedron must conform to the properties.4.2 Implicit from the terminology, √ d ij ≥ 0 ⇔ d ij ≥ 0 is always assumed.4.3 What we call self-distance, Marsden calls nondegeneracy. [160,1.6] Kreyszig callsthese first metric properties axioms of the metric; [140, p.4] Blumenthal refers to them aspostulates. [32, p.15]