v2010.10.26 - Convex Optimization

5.4. EDM DEFINITION 413Figure 120: Sphere-packing illustration from [373, kissing number].Translucent balls illustrated all have the same diameter.where, for A i ∈ R 6×6 (919)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 )(928)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 (921). 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 (915).25.4.2.3.4 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? [394] 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)