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

Published 22 Aug 2018 in math.MG and math.CO | (1808.07299v2)

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.

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