v2010.10.26 - Convex Optimization

5.4. EDM DEFINITION 415is found from the maximal number of spheres N that solve this semidefiniteprogram in a given affine dimension r . Matrix W can be interpreted as thedirection of search through the positive semidefinite cone S N−1+ for a rank-roptimal solution −VN TD⋆ V N ; it is constant, in this program, determined bya method disclosed in4.4.1. In two dimensions,⎡W =⎢⎣4 1 2 −1 −1 11 4 −1 −1 2 12 −1 4 1 1 −1−1 −1 1 4 1 2−1 2 1 1 4 −11 1 −1 2 −1 4⎤1⎥6⎦(931)In three dimensions,⎡W =⎢⎣9 1 −2 −1 3 −1 −1 1 2 1 −2 11 9 3 −1 −1 1 1 −2 1 2 −1 −1−2 3 9 1 2 −1 −1 2 −1 −1 1 2−1 −1 1 9 1 −1 1 −1 3 2 −1 13 −1 2 1 9 1 1 −1 −1 −1 1 −1−1 1 −1 −1 1 9 2 −1 2 −1 2 3−1 1 −1 1 1 2 9 3 −1 1 −2 −11 −2 2 −1 −1 −1 3 9 2 −1 1 12 1 −1 3 −1 2 −1 2 9 −1 1 −11 2 −1 2 −1 −1 1 −1 −1 9 3 1−2 −1 1 −1 1 2 −2 1 1 3 9 −11 −1 2 1 −1 3 −1 1 −1 1 −1 9⎤112⎥⎦(932)A four-dimensional solution has rational direction matrix as well, but thesedirection matrices are not unique and their precision not critical. Here is anoptimal point list 5.14 in Matlab output format:5.14 An optimal five-dimensional point list is known: The answer was known at least 175years ago. I believe Gauss knew it. Moreover, Korkine & Zolotarev proved in 1882 that D 5is the densest lattice in five dimensions. So they proved that if a kissing arrangement infive dimensions can be extended to some lattice, then k(5)= 40. Of course, the conjecturein the general case also is: k(5)= 40. You would like to see coordinates? Easily.Let A= √ 2. Then p(1)=(A,A,0,0,0), p(2)=(−A,A,0,0,0), p(3)=(A, −A,0,0,0), ...p(40)=(0,0,0, −A, −A); i.e., we are considering points with coordinates that have twoA and three 0 with any choice of signs and any ordering of the coordinates; the same