# The lattice packing problem in dimension 9 by Voronoi's algorithm

## TL;DR

This paper computes all perfect lattices in dimension 9 using Voronoi's algorithm, establishing the densest packing, the Hermite constant, and the set of possible kissing numbers in that dimension.

## Contribution

It completes the enumeration of perfect lattices in dimension 9, a task previously limited to dimension 8, and derives new results on lattice density and kissing numbers.

## Key findings

- The laminated lattice Λ₉ is the densest in dimension 9.
- The Hermite constant γ₉ equals 2 in dimension 9.
- The set of possible kissing numbers in dimension 9 is explicitly characterized.

## Abstract

In 1908, Voronoi introduced an algorithm that solves the lattice packing problem in any dimension in finite time. Voronoi showed that any lattice with optimal packing density must be a so-called perfect lattice, and his algorithm enumerates the finitely many perfect lattices up to similarity in a fixed dimension. However, due to the high complexity of the algorithm this enumeration had, until now, only been completed up to dimension 8. In this work we compute all 2237251040 perfect lattices in dimension 9 via Voronoi's algorithm. As a corollary, this shows that the laminated lattice $\Lambda_9$ gives the densest lattice packing in dimension 9. Equivalently, we show that the Hermite constant $\gamma_9$ in dimension 9 equals $2$. Furthermore, we extend a result by Watson (1971) and show that the set of possible kissing numbers in dimension 9 is precisely $2 \cdot \{ 1, \ldots, 91, 99, 120, \ldots, 129, 136 \}$.