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

  • Fernando Mário de Oliveira Filho (a1) and Frank Vallentin (a2)

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.

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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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1. Ball, K., An elementary introduction to modern convex geometry. In Flavors of Geometry (Mathematical Sciences Research Institute Publications 31 ), Cambridge University Press (Cambridge, 1997), 158.
2. Blum, A., Hopcroft, J. and Kannan, R., Foundations of Data Science, 2018, http://www.cs.cornell.edu/jeh.
3. Croft, H. T., Falconer, K. J. and Guy, R., Unsolved Problems in Geometry, Springer (New York, 1991).
4. DeCorte, E., de Oliveira Filho, F. M. and Vallentin, F., Complete positivity and distance-avoiding sets. Preprint, 2018, arXiv:1804:09099.
5. Kalai, G., Some old and new problems in combinatorial geometry I: around Borsuk’s problem. In Surveys in Combinatorics 2015 (London Mathematical Society Lecture Note Series 424 ), Cambridge University Press (Cambridge, 2015), 147174.
6. Larman, D. G. and Rogers, C. A., The realization of distances within sets in Euclidean space. Mathematika 19 1972, 124.
7. Matoušek, J., Lectures on Discrete Geometry (Graduate Texts in Mathematics 212 ), Springer (New York, 2002).
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A COUNTEREXAMPLE TO A CONJECTURE OF LARMAN AND ROGERS ON SETS AVOIDING DISTANCE 1

  • Fernando Mário de Oliveira Filho (a1) and Frank Vallentin (a2)

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