# The sphere packing problem in dimension 8The sphere packing problem in dimension 8

@article{Viazovska2017TheSP, title={The sphere packing problem in dimension 8The sphere packing problem in dimension 8}, author={Maryna S. Viazovska}, journal={Annals of Mathematics}, year={2017}, volume={185}, pages={991-1015} }

In this paper we prove that no packing of unit balls in Euclidean space R-8 has density greater than that of the E8-lattice packing.

## 106 Citations

The sphere packing problem in dimension 24

- Mathematics
- 2016

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…

Towards a proof of the 24-cell conjecture

- Mathematics
- 2017

This review paper is devoted to the problems of sphere packings in 4 dimensions. The main goal is to find reasonable approaches for solutions to problems related to densest sphere packings in…

Compact Packings of Space with Two Sizes of Spheres

- Computer Science, MathematicsDiscret. Comput. Geom.
- 2021

It is shown that the compact packings of Euclidean three-dimensional space with two sizes of spheres are exactly those obtained by filling with spheres of size 2-1, and the octahedral holes of a close-packing of sphere of size 1.

Sphere packings, Lattices and Codes

- 2021

1. The sphere packing problem. Statement of the problem. Definition and basic properties of lattices: fundamental region, discriminant, Gram matrix. Density of a lattice packing and of a general…

Bounds for several-disk packings of hyperbolic surfaces

- MathematicsJournal of Topology and Analysis
- 2018

For any given [Formula: see text], this paper gives upper bounds on the radius of a packing of a complete hyperbolic surface of finite area by [Formula: see text] equal-radius disks in terms of the…

Efficient Approximations for the Online Dispersion Problem

- Mathematics, Computer ScienceSIAM J. Comput.
- 2019

The dispersion problem has been widely studied in computational geometry and facility location and is closely related to the packing problem. The goal is to locate $n$ points (e.g., facilities or p...

Exponential improvements for superball packing upper bounds

- Mathematics
- 2020

Abstract We prove that for all fixed p > 2 , the translative packing density of unit l p -balls in R n is at most 2 ( γ p + o ( 1 ) ) n with γ p − 1 / p . This is the first exponential improvement in…

Locally Optimal 2-Periodic Sphere Packings

- Mathematics, Computer ScienceDiscret. Comput. Geom.
- 2020

This work generalizes Voronoi’s method to m > 1 and presents a procedure to enumerate all locally optimal 2-periodic sphere packings in any dimension, provided there are finitely many.

Coloring the Voronoi tessellation of lattices

- MathematicsJournal of the London Mathematical Society
- 2019

In this paper we define the chromatic number of a lattice: It is the least number of colors one needs to color the interiors of the cells of the Voronoi tessellation of a lattice so that no two cells…

Computing Upper Bounds for the Packing Density of Congruent Copies of a Convex Body

- Mathematics
- 2018

In this paper we prove a theorem that provides an upper bound for the density of packings of congruent copies of a given convex body in \(\mathbb {R}^n\); this theorem is a generalization of the…

## References

SHOWING 1-10 OF 22 REFERENCES

New upper bounds on sphere packings I

- Mathematics
- 2003

We continue the study of the linear programming bounds for sphere packing introduced by Cohn and Elkies. We use theta series to give another proof of the principal theorem, and present some related…

Kissing numbers, sphere packings, and some unexpected proofs

- 2004

The “kissing number problem” asks for the maximal number of white spheres that can touch a black sphere of the same size in n-dimensional space. The answers in dimensions one, two and three are…

What are all the best sphere packings in low dimensions?

- Mathematics, Computer ScienceDiscret. Comput. Geom.
- 1995

We describe what may beall the best packings of nonoverlapping equal spheres in dimensionsn ≤10, where “best” means both having the highest density and not permitting any local improvement. For…

Optimal asymptotic bounds for spherical designs

- Mathematics
- 2010

In this paper we prove the conjecture of Korevaar and Meyers: for each $N\ge c_dt^d$ there exists a spherical $t$-design in the sphere $S^d$ consisting of $N$ points, where $c_d$ is a constant…

Universally optimal distribution of points on spheres

- Mathematics
- 2006

We study configurations of points on the unit sphere that minimize potential energy for a broad class of potential functions (viewed as functions of the squared Euclidean distance between points).…

A First Course in Modular Forms

- Mathematics
- 2008

Modular Forms, Elliptic Curves, and Modular Curves.- Modular Curves as Riemann Surfaces.- Dimension Formulas.- Eisenstein Series.- Hecke Operators.- Jacobians and Abelian Varieties.- Modular Curves…

Spherical codes and designs

- Mathematics
- 1977

Publisher Summary This chapter provides an overview of spherical codes and designs. A finite non-empty set X of unit vectors in Euclidean space R d has several characteristics, such as the dimension…

Bounds for unrestricted codes, by linear programming

- Mathematics
- 1972

The paper describes a problem of linear programming associated with distance properties of unrestricted codes. As a solution to the problem, one obtains an .upper bound for the number of words in…

Elliptic modular forms and their applications.

- Computer Science
- 2008

These notes give a brief introduction to a number of topics in the classical theory of modular forms, based on various courses held at the College de France in the years 2000–2004.

Borcherds Products on O(2,l) and Chern Classes of Heegner Divisors

- Mathematics
- 2002

Introduction.- Vector valued modular forms for the metaplectic group. The Weil representation. Poincare series and Einstein series. Non-holomorphic Poincare series of negative weight.- The…