Swanepoel, Konrad ORCID: 0000-0002-1668-887X
(2020)
*Favourite distances in 3-space.*
Electronic Journal of Combinatorics, 27 (2).
pp. 1-11.
ISSN 1077-8926

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## Abstract

Let S be a set of n points in Euclidean 3-space. Assign to each x ∈ S a distance r(x) > 0, and let er(x,S) denote the number of points in S at distance r(x) from x. Avis, Erdo ̋s and Pach (1988) introduced the extremal quantity f3(n) = max ﰝx∈S er(x, S), where the maximum is taken over all n-point subsets S of 3-space and all assignments r: S → (0,∞) of distances. We show that if the pair (S,r) maximises f3(n) and n is sufficiently large, then, except for at most 2 points, S is contained in a circle C and the axis of symmetry L of C, and r(x) equals the distance from x to C for each x ∈ S ∩ L. This, together with a new construction, implies that f3(n) = n2/4 + 5n/2 + O(1).

Item Type: | Article |
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Official URL: | https://www.combinatorics.org/ |

Additional Information: | © 2020 The Author |

Divisions: | Mathematics |

Subjects: | Q Science > QA Mathematics |

Date Deposited: | 28 Apr 2020 11:57 |

Last Modified: | 20 Oct 2021 01:02 |

URI: | http://eprints.lse.ac.uk/id/eprint/104165 |

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