Optimality and uniqueness of the Leech lattice among lattices

By Henry Cohn, Abhinav Kumar. 2004

153 citations recorded by the source corpus.

We prove that the Leech lattice is the unique densest lattice in R^24. The proof combines human reasoning with computer verification of the properties of certain explicit polynomials. We furthermore prove that no sphere packing in R^24 can exceed the Leech lattice's density by a factor of more than 1+1.65*10^(-30), and we give a new proof that E_8 is the unique densest lattice in R^8.

Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/968c99a07543a0da080c4f20aaa5d67fc9e0801b

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Theory and Mathematics

1. Salvatore Torquato
2. Frank H. Stillinger
3. Stephanie Vance
4. Günter M. Ziegler
5. Henry Cohn
6. Jirí Matousek

In Discrete Mathematics and Combinatorics

1. Salvatore Torquato
2. Frank H. Stillinger
3. Stephanie Vance
4. Günter M. Ziegler
5. Henry Cohn
6. Jirí Matousek
7. Fernando Muro

In Numerical Analysis

1. Salvatore Torquato
2. Danylo V. Radchenko
3. Frank Vallentin
4. Henry Cohn
5. Maryna S. Viazovska
6. Frank H. Stillinger
7. Abhinav Kumar
8. Stephen D. Miller

Citations in the corpus

• Classification of eight dimensional perfect forms (2006, background): “A completely different, computer assisted proof is in [CoKu04].”
• Kissing numbers, sphere packings, and some unexpected proofs (2004, background): “The best currently known function was found by Cohn and Kumar [6], a radial function that consists of a polynomial of degree 803 (evaluated at |x|2, so technically it is a polynomial of degree 1606) multiplied by e−π|x|2 .”
• A conceptual breakthrough in sphere packing (2016, background): “It was understood by a number of people within a few days of her arXiv posting, and within a week it led to further progress: Abhinav Kumar, Stephen D. Miller, Danylo Radchenko, and I worked with Viazovska to adapt her methods to prove that the Leech lattice is an optimal sphere packing in twenty-fo”
• Universally optimal distribution of points on spheres (2006, unknown): “Perhaps one could combine their theorem with the techniques in this paper and [CK1] to prove that E 8 and Λ 24 are the global minima among lattices in their dimensions.”
• Improved sphere packing lower bounds from Hurwitz lattices (2011, background): “ basis for Λ.) Unfortunately, even with this explicit density formula the lattice sphere packing density ∆n,L is unknown for all dimensions n > 8, with the exception of dimension 24; see [CS2] and [CK] for more details regarding the known optimal lattices. Despite the absence of a known general s”
• Jammed hard-particle packings: From Kepler to Bernal and beyond (2010, methodology): “The remarkably symmetric E8 and Leech lattices in R8 and R24, respectively, are most likely the densest packings in these dimensions (Cohn and Kumar, 2009).”
• The sphere packing problem in dimension 24 (2016, background): “The optimality and uniqueness of the Leech lattice were previously known only among lattice packings [3], which is a far more restrictive setting.”