Complete positivity and distance-avoiding sets

TL;DR

This paper introduces a new mathematical framework using completely-positive functions to characterize and optimize maximum-density distance-avoiding sets, leading to improved results in geometric combinatorics.

Contribution

It defines the cone of completely-positive functions and applies it to characterize maximum-density distance-avoiding sets via convex optimization, advancing the theoretical understanding.

Findings

Reproves existing results on distance-avoiding sets

Improves bounds and characterizations for sets on the sphere and Euclidean space

Provides a new convex optimization approach for these problems

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.