v2007.09.13 - Convex Optimization

5.14. FIFTH PROPERTY OF EUCLIDEAN METRIC 3815.14.3 Path not followedAs a means to test for realizability of four or more points, anintuitively appealing way to augment the four Euclidean metric propertiesis to recognize generalizations of the triangle inequality: In thecase N = 4, the three-dimensional analogue to triangle & distance istetrahedron & facet-area, while in the case N = 5 the four-dimensionalanalogue is polychoron & facet-volume, ad infinitum. For N points, N + 1metric properties are required.5.14.3.1 N = 4Each of the four facets of a general tetrahedron is a triangle and itsrelative interior. Suppose we identify each facet of the tetrahedron by itsarea-squared: c 1 , c 2 , c 3 , c 4 . Then analogous to metric property 4, we maywrite a tight 5.54 area inequality for the facets√ci ≤ √ c j + √ c k + √ c l , i≠j ≠k≠l∈{1, 2, 3, 4} (968)which is a generalized “triangle” inequality [165,1.1] that follows from√ci = √ c j cos ϕ ij + √ c k cos ϕ ik + √ c l cos ϕ il (969)[176] [279, Law of Cosines] where ϕ ij is the dihedral angle at the commonedge between triangular facets i and j .If D is the EDM corresponding to the whole tetrahedron, thenarea-squared of the i th triangular facet has a convenient formula in termsof D i ∈ EDM N−1 the EDM corresponding to that particular facet: From theCayley-Menger determinant 5.55 for simplices, [279] [86] [111,4] [60,3.3]the i th facet area-squared for i∈{1... N} is (A.4.1)5.54 The upper bound is met when all angles in (969) are simultaneously 0; that occurs,for example, if one point is relatively interior to the convex hull of the three remaining.5.55 whose foremost characteristic is: the determinant vanishes [ if and ] only if affine0 1Tdimension does not equal penultimate cardinality; id est, det = 0 ⇔ r < N −11 −Dwhere D is any EDM (5.7.3.0.1). Otherwise, the determinant is negative.