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

For $n\ge 2$ 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 $(1\mathrm{/}2{)}^n$ times the volume of the ball. This disproves a conjecture of Larman and Rogers from 1972.