v2010.10.26 - Convex Optimization

492 CHAPTER 5. EUCLIDEAN DISTANCE MATRIXc i ===[ ]−1 0 12 N−2 (N −2)! 2 det T1 −D i(−1) N2 N−2 (N −2)! 2 detD i(1167)(1 T D −1i 1 ) (1168)(−1) N2 N−2 (N −2)! 2 1T cof(D i ) T 1 (1169)where D i is the i th principal N −1×N −1 submatrix 5.62 of D ∈ EDM N ,and cof(D i ) is the N −1×N −1 matrix of cofactors [331,4] correspondingto D i . The number of principal 3 × 3 submatrices in D is, of course, equalto the number of triangular facets in the tetrahedron; four (N!/(3!(N −3)!))when N = 4.5.14.3.1.1 Exercise. Sufficiency conditions for an EDM of four points.Triangle inequality (property 4) and area inequality (1165) are conditionsnecessary for D to be an EDM. Prove their sufficiency in conjunction withthe remaining three Euclidean metric properties.5.14.3.2 N = 5Moving to the next level, we might encounter a Euclidean body calledpolychoron: a bounded polyhedron in four dimensions. 5.63 Our polychoronhas five (N!/(4!(N −4)!)) facets, each of them a general tetrahedron whosevolume-square c i is calculated using the same formula; (1167) whereD is the EDM corresponding to the polychoron, and D i is the EDMcorresponding to the i th facet (the principal 4 × 4 submatrix of D ∈ EDM Ncorresponding to the i th tetrahedron). The analogue to triangle & distanceis now polychoron & facet-volume. We could then write another generalized“triangle” inequality like (1165) but in terms of facet volume; [379,IV]√ci ≤ √ c j + √ c k + √ c l + √ c m , i≠j ≠k≠l≠m∈{1... 5} (1170)5.62 Every principal submatrix of an EDM remains an EDM. [235,4.1]5.63 The simplest polychoron is called a pentatope [373]; a regular simplex hence convex.(A pentahedron is a three-dimensional body having five vertices.)