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.


Henry Cohn

Microsoft Research New England, One Memorial Drive, Cambridge, MA 02142, USA

Abhinav Kumar

Department of Mathematics, Stony Brook University, Stony Brook, NY 11794, USA

Stephen D. Miller

Department of Mathematics, Hill Center--Busch Campus, Rutgers University, 110 Frelinghuysen Rd., Piscataway, NY 08854-8019, USA

Danylo Radchenko

Max Planck Institute for Mathematics, Bonn, Germany

Current address:

The Abdus Salam International Centre for Theoretical Physics, Str. Costiera 11, 34151 Trieste, Italy Maryna Viazovska

Berlin Mathematical School, Str. des 17. Juni 136, 10623 Berlin, Germany, and
Humboldt University of Berlin, Rudower Chaussee 25, 12489 Berlin, Germany