v2010.10.26 - Convex Optimization

5.14. FIFTH PROPERTY OF EUCLIDEAN METRIC 4915.14.3 Path not followedAs a means to test for realizability of four or more points, an intuitivelyappealing way to augment the four Euclidean metric properties is torecognize generalizations of the triangle inequality: In the case ofcardinality N = 4, the three-dimensional analogue to triangle & distance istetrahedron & facet-area, while in case N = 5 the four-dimensional analogueis polychoron & facet-volume, ad infinitum. For N points, N + 1 metricproperties are required.5.14.3.1 N = 4Each of the four facets of a general tetrahedron is a triangle and its relativeinterior. Suppose we identify each facet of the tetrahedron by its area-square:c 1 , c 2 , c 3 , c 4 . Then analogous to metric property 4, we may write a tight 5.60area inequality for the facets√ci ≤ √ c j + √ c k + √ c l , i≠j ≠k≠l∈{1, 2, 3, 4} (1165)which is a generalized “triangle” inequality [227,1.1] that follows from√ci = √ c j cos ϕ ij + √ c k cos ϕ ik + √ c l cos ϕ il (1166)[242] [373, 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, then area-squareof the i th triangular facet has a convenient formula in terms of D i ∈ EDM N−1the EDM corresponding to that particular facet: From the Cayley-Mengerdeterminant 5.61 for simplices, [373] [131] [162,4] [88,3.3] the i th facetarea-square for i∈{1... N} is (A.4.1)5.60 The upper bound is met when all angles in (1166) are simultaneously 0; that occurs,for example, if one point is relatively interior to the convex hull of the three remaining.5.61 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.