The sphere packing problem in dimension 24
Henry Cohn, Abhinav Kumar, Stephen D. Miller, Danylo Radchenko, Maryna, Viazovska
TL;DR
This paper proves that the Leech lattice provides the densest sphere packing in 24 dimensions, extending Viazovska's breakthrough in eight dimensions, and establishes its uniqueness as the optimal periodic packing.
Contribution
It introduces an optimal auxiliary function for linear programming bounds in 24 dimensions, confirming the Leech lattice's optimality and uniqueness.
Findings
Leech lattice is the densest packing in 24 dimensions.
Leech lattice is the unique optimal periodic packing.
An optimal auxiliary function analogous to Viazovska's was found.
Abstract
Building on Viazovska's recent solution of the sphere packing problem in eight dimensions, we prove that the Leech lattice is the densest packing of congruent spheres in twenty-four dimensions and that it is the unique optimal periodic packing. In particular, we find an optimal auxiliary function for the linear programming bounds, which is an analogue of Viazovska's function for the eight-dimensional case.
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