New upper bounds on sphere packings II

By Henry Cohn. 2001

71 citations recorded by the source corpus.

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 results and conjectures.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/574abd37fbc9a41fc61415d9f33e7dd57a7aac20

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In Applied Mathematics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Optimality and uniqueness of the Leech lattice among lattices (2004, unknown): “heorem 11.7 (Blichfeldt, Vetˇcinkin). The E8 root lattice is the unique densest lattice in R8, up to scaling and isometries of R8. Our work is motivated by the paper [CE] by Cohn and Elkies (see also [Co]), which proves upper bounds for the sphere packing density. In particular, the main theorem in ”
Convolution roots of radial positive definite functions with compact support (2004, background): “Note the similarity of the minimizing density (52) to the solution of the extremal problem in Section 3 of [ 15 ] and Section 6 of [16], with its interesting and surprising connections to the sphere packing problem.”
Jammed hard-particle packings: From Kepler to Bernal and beyond (2010, background): “There has been resurgent interest in sphere packings for d > 3 in both the physical and mathematical sciences (Adda-Bedia et al., 2008; Cohn, 2002; Cohn and Elkies, 2003; Cohn and Kumar, 2009; Conway and Sloane, 1995; Elkies, 2000; Frisch and Percus, 1999; Lue et al., 2010; van Meel et al., 2009;…”
Universally optimal distribution of points on spheres (2006, unknown): “This proposition is closely analogous to the linear programming bounds for sphere packing introduced in [CE] (see also [ Co , CK1, CK2]).”
The densest lattice in twenty-four dimensions (2004, methodology): “We will apply it using a function f such that f(x) ≤ 0 for |x| suﬃciently large, but fb(t) ≥ 0 for all t. This technique was used in [CE] to prove upper bounds on the sphere packing density (see also [C]). These bounds appear to be sharp in R8 and R24, which would solve the full sphere packing probl”