The decomposition of the hypermetric cone into L-domains

TL;DR

This paper investigates how the hypermetric cone, which parameterizes Delaunay polytopes in lattices, decomposes into L-domains, revealing a precise count of principal domains and detailed structures for low dimensions.

Contribution

It proves the exact number of principal L-domains within the hypermetric cone and provides detailed decompositions for dimensions up to 4, with computational results for dimension 5.

Findings

The hypermetric cone contains exactly (1/2)n! principal L-domains.

Detailed decompositions of the cone for n=2,3,4 are provided.

Computational results for n=5 reveal complex structure influenced by root system D4.

Abstract

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 $\HYP_5$.