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

Fernando M\'ario de Oliveira Filho; Frank Vallentin

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

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arxiv 1808.07299 v2 pith:OUU636FW submitted 2018-08-22 math.MG math.CO

Fernando Mário de Oliveira Filho , Frank Vallentin This is my paper · ORCID

classification math.MGmath.CO

keywords ballconjecturedistancelarmanrogersvolumeavoidingconstruct

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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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