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A counterexample to a conjecture of Larman and Rogers on sets avoiding distance 1. (English) Zbl 1433.52019

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.].

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

Citations:

Zbl 0246.05020
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References:

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[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
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