# The sphere packing problem in dimension $24$

### 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.

Computer code for verifying the calculations in this paper is contained in the ancillary file annals.2017.185-3.p08.appendix.txt.

## Authors

Henry Cohn

Microsoft Research New England, Cambridge, MA

Abhinav Kumar

Stony Brook University, Stony Brook, NY

Stephen D. Miller

Rutgers University, Piscataway, NJ