Complete positivity and distance-avoiding sets

Abstract

We introduce the cone of completely-positive functions, a subset of the cone of positive-type functions, and use it to fully characterize maximum-density distance-avoiding sets as the optimal solutions of a convex optimization problem. As a consequence of this characterization, it is possible to reprove and improve many results concerning distance-avoiding sets on the sphere and in Euclidean space.

57 pages; final version published at Math. Program

References in corpus (5)

• The sphere packing problem in dimension 24
• The sphere packing problem in dimension 8
• New upper bounds for kissing numbers from semidefinite programming
• Improved sphere packing lower bounds from Hurwitz lattices
• A counterexample to a conjecture of Larman and Rogers on sets avoiding distance 1

Cited by in corpus (6)

• Density estimates of 1-avoiding sets via higher order correlations
• The density of planar sets avoiding unit distances
• Positive semidefinite approximations to the cone of copositive kernels
• A recursive Lovász theta number for simplex-avoiding sets
• On the support of a non-autocorrelated function on a hyperbolic surface
• On the integrality gap of the maximum-cut semidefinite programming relaxation in fixed dimension