v2007.09.17 - Convex Optimization

5.4. EDM DEFINITION 307ten affine equality constraints in all on a Gram matrix G∈ S 6 (731). Let’srealize this as a convex feasibility problem (with constraints written in thesame order) also assuming 0 geometric center (730):findD∈S 6 h−V DV 1 2 ∈ S6subject to tr ( )D(e i e T j + e j e T i ) 1 2 = l2ij , j−1 = (i = 1... 6) mod 6tr ( − 1V DV (A 2 i + A T i ) 2) 1 = cos ϕi , i = 1, 2, 3tr(− 1 2 V DV ) = 1−V DV ≽ 0 (743)where, for A i ∈ R 6×6 (738)A 1 = (e 1 − e 6 )(e 3 − e 4 ) T /(l 61 l 34 )A 2 = (e 2 − e 1 )(e 4 − e 5 ) T /(l 12 l 45 )A 3 = (e 3 − e 2 )(e 5 − e 6 ) T /(l 23 l 56 )(744)and where the first constraint on length-square l 2 ij can be equivalently writtenas a constraint on the Gram matrix −V DV 1 via (740). We show how to2numerically solve such a problem by alternating projection inE.10.2.1.1.Barvinok’s Proposition 2.9.3.0.1 asserts existence of a list, correspondingto Gram matrix G solving this feasibility problem, whose affine dimension(5.7.1.1) does not exceed 3 because the convex feasible set is bounded bythe third constraint tr(− 1 V DV ) = 1 (734).25.4.2.2.3 Example. Kissing-number of sphere packing.Two nonoverlapping Euclidean balls are said to kiss if they touch. Anelementary geometrical problem can be posed: Given hyperspheres, eachhaving the same diameter 1, how many hyperspheres can simultaneouslykiss one central hypersphere? [302] The noncentral hyperspheres are allowed,but not required, to kiss.As posed, the problem seeks the maximal number of spheres K kissinga central sphere in a particular dimension. The total number of spheres isN = K + 1. In one dimension the answer to this kissing problem is 2. In twodimensions, 6. (Figure 7)The question was presented in three dimensions to Isaac Newton in thecontext of celestial mechanics, and became controversy with David Gregoryon the campus of Cambridge University in 1694. Newton correctly identified