The sphere packing problem in dimension $24$


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.


Henry Cohn

Microsoft Research New England, Cambridge, MA

Abhinav Kumar

Stony Brook University, Stony Brook, NY

Stephen D. Miller

Rutgers University, Piscataway, NJ

Danylo Radchenko

Max Planck Institute for Mathematics, Bonn, Germany

Current address:

The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy Maryna Viazovska

Berlin Mathematical School and Humboldt University of Berlin, Berlin, Germany

Current address:

École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland