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Favourite distances in 3-space

Swanepoel, Konrad ORCID: 0000-0002-1668-887X (2020) Favourite distances in 3-space. Electronic Journal of Combinatorics. ISSN 1077-8926 (In Press)

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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
Official URL:
Additional Information: © 2020 The Author
Divisions: Mathematics
Subjects: Q Science > QA Mathematics
Date Deposited: 28 Apr 2020 11:57
Last Modified: 28 Jun 2020 23:31

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