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

Fernando Mário de Oliveira Filho; Frank Vallentin

doi:10.1112/S0025579319000160

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

Metric Geometry 2019-05-15 v2 Combinatorics

Authors: Fernando Mário de Oliveira Filho , Frank Vallentin

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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.

Keywords

distance avoiding sets unit ball measure counterexample larman-rogers conjecture

Cite

@article{arxiv.1808.07299,
  title  = {A counterexample to a conjecture of Larman and Rogers on sets avoiding distance 1},
  author = {Fernando Mário de Oliveira Filho and Frank Vallentin},
  journal= {arXiv preprint arXiv:1808.07299},
  year   = {2019}
}

Comments

3 pages, 1 figure; final version to appear in Mathematika

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

Abstract

Keywords

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