De Oliveira Filho, Fernando Mário; Vallentin, Frank A counterexample to a conjecture of Larman and Rogers on sets avoiding distance 1. (English) Zbl 1433.52019 Mathematika 65, No. 3, 785-787 (2019). D. G. Larman and C. A. Rogers conjectured in [ibid. 19, 1–24 (1972; Zbl 0246.05020)] the following: “Suppose that the distance 1 is not realized in a closed subset \(S\) of a spherical ball \(B\) of radius 1. Then the Lebesgue measure of \(S\) is less than \((1/2)^n\) times the Lebesgue measure of \(B\).” In the present paper, the authors construct, for each \(n \geq 2\), 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 the aforementioned conjecture of Larman and Rogers from [loc. cit.]. Reviewer: Bogdan Suceavă (Fullerton) Cited in 1 Document MSC: 52C10 Erdős problems and related topics of discrete geometry 51K99 Distance geometry 28A75 Length, area, volume, other geometric measure theory 28A12 Contents, measures, outer measures, capacities Keywords:Lebesgue measure; Larman-Rogers conjecture; Moser conjecture Citations:Zbl 0246.05020 PDFBibTeX XMLCite \textit{F. M. De Oliveira Filho} and \textit{F. Vallentin}, Mathematika 65, No. 3, 785--787 (2019; Zbl 1433.52019) Full Text: DOI arXiv References: [1] Ball, K., An elementary introduction to modern convex geometry. In Flavors of Geometry, Cambridge University Press (Cambridge, 1997), 1-58. · Zbl 0901.52002 [2] Blum, A., Hopcroft, J. and Kannan, R., Foundations of Data Science, 2018, http://www.cs.cornell.edu/jeh. · Zbl 1477.68002 [3] Croft, H. T., Falconer, K. J. and Guy, R., Unsolved Problems in Geometry, Springer (New York, 1991). · Zbl 0748.52001 [4] Decorte, E., De Oliveira Filho, F. M. and Vallentin, F., Complete positivity and distance-avoiding sets. Preprint, 2018, arXiv:1804:09099. · Zbl 1495.46060 [5] Kalai, G., Some old and new problems in combinatorial geometry I: around Borsuk’s problem. In Surveys in Combinatorics 2015, Cambridge University Press (Cambridge, 2015), 147-174. · Zbl 1361.51008 [6] Larman, D. G. and Rogers, C. A., The realization of distances within sets in Euclidean space. Mathematika191972, 1-24. · Zbl 0246.05020 [7] Matoušek, J., Lectures on Discrete Geometry, (2002), Springer: Springer, New York · Zbl 0999.52006 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.