A New Algorithm in Geometry of Numbers

A lattice Delaunay polytope P is called perfect if its Delaunay sphere is the only ellipsoid circumscribed about P. We present a new algorithm for finding perfect Delaunay polytopes. Our method overcomes the major shortcomings of the previously used method. We have implemented and used our algorithm for finding perfect Delaunay polytopes in dimensions 6, 7, 8. Our findings lead to a new conjecture that sheds light on the structure of lattice Delaunay tilings.Comment: 7 pages, 3 figures; Proceedings of ISVD-07, International Symposium on Voronoi diagrams in Science and Engineering held in July of 2007 in Wales, U