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A COUNTEREXAMPLE TO A CONJECTURE OF LARMAN AND ROGERS ON SETS AVOIDING DISTANCE 1

Part of: Discrete geometry Distance geometry

Published online by Cambridge University Press:  14 May 2019

Fernando Mário de Oliveira Filho  and
Frank Vallentin
Fernando Mário de Oliveira Filho
Affiliation:
Delft Institute of Applied Mathematics, Delft University of Technology, Van Mourik Broekmanweg 6, 2628 XE Delft, The Netherlands email fmario@gmail.com
Frank Vallentin
Affiliation:
Mathematisches Institut, Universität zu Köln, Weyertal 86–90, 50931 Köln, Germany email frank.vallentin@uni-koeln.de
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Abstract

For each $n\geqslant 2$ 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 unit ball. This disproves a conjecture of Larman and Rogers from 1972.

MSC classification

Primary: 52C10: Erdős problems and related topics of discrete geometry 51K99: None of the above, but in this section
Type
Research Article
Information
Mathematika , Volume 65 , Issue 3 , 2019 , pp. 785 - 787
Copyright
Copyright © University College London 2019 

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Footnotes

The second author was partially supported by the SFB/TRR 191 “Symplectic Structures in Geometry, Algebra and Dynamics”, funded by the DFG, and the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie agreement number 764759.

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