The decomposition of the hypermetric cone into L-domains

The hypermetric cone \HYP_{n+1} is the parameter space of basic Delaunay polytopes in n-dimensional lattice. The cone \HYP_{n+1} is polyhedral; one way of seeing this is that modulo image by the covariance map \HYP_{n+1} is a finite union of L-domains, i.e., of parameter space of full Delaunay tessellations. In this paper, we study this partition of the hypermetric cone into L-domains. In particular, it is proved that the cone \HYP_{n+1} of hypermetrics on n+1 points contains exactly {1/2}n! principal L-domains. We give a detailed description of the decomposition of \HYP_{n+1} for n=2,3,4 and a computer result for n=5 (see Table \ref{TableDataHYPn}). Remarkable properties of the root system $\mathsf{D}_4$ are key for the decomposition of \HYP_5.Comment: 20 pages 2 figures, 2 table

Graphs 4n4_n that are isometrically embeddable in hypercubes

A connected 3-valent plane graph, whose faces are $q$- or 6-gons only, is called a {\em graph $q_n$}. We classify all graphs $4_n$, which are isometric subgraphs of a $m$-hypercube $H_m$.Comment: 18 pages, 25 drawing

Classification of eight dimensional perfect forms

In this paper, we classify the perfect lattices in dimension 8. There are 10916 of them. Our classification heavily relies on exploiting symmetry in polyhedral computations. Here we describe algorithms making the classification possible.Comment: 14 page