A counterexample to a conjecture of Larman and Rogers on sets avoiding distance 1

TL;DR

This paper constructs a measurable subset of the unit ball in high-dimensional space that avoids pairs of points at distance 1 and has a volume larger than previously conjectured, disproving a longstanding conjecture.

Contribution

It provides a counterexample to Larman and Rogers' 1972 conjecture about the maximum volume of sets avoiding distance 1.

Findings

Constructed a measurable set with volume > (1/2)^n of the unit ball

Disproved the conjecture by Larman and Rogers from 1972

Showed existence of larger sets avoiding distance 1 in high dimensions

Abstract

For $n \geq 2$ we construct a measurable subset of the unit ball in $\mathbb{R}^n$ that does not contain pairs of points at distance 1 and whose volume is greater than $(1/2)^n$ times the volume of the ball. This disproves a conjecture of Larman and Rogers from 1972.