v2004.06.19 - Convex Optimization

4.2. METRIC REQUIREMENTS 131correspond to D in (323). Such a list is not unique because any rotation,reflection, or translation (§4.5) of the points in Figure 4.1 would produce thesame EDM D .4.2 Metric requirementsFor i,j = 1... N , the Euclidean distance between points x i and x j mustsatisfy the axiomatic requirements imposed by any metric space: [38, §1.1][36, §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 axioms: specifically, each entrymust be nonnegative, 4.2 the main diagonal must be 0 , 4.3 an EDM must besymmetric. The fourth axiom provides upper and lower bounds for eachentry. Axiom 4 is true more generally when there are no restrictions onindices i,j,k , but furnishes no new information.4.3 ∃ fifth Euclidean axiomThe four axioms of the Euclidean metric provide information insufficient tocertify that a bounded convex polyhedron more complicated than a trianglehas a Euclidean realization. [84, §2] Yet any list of points or the vertices ofany bounded convex polyhedron must conform to the axioms.4.2 Implicit from the terminology, √ d ij ≥ 0 ⇔ d ij ≥ 0 is always assumed.4.3 What we call zero self-distance, Marsden calls nondegeneracy. [36, §1.6]