Furthest neighbours in space
Let \(X = \{x_1, \dots, x_n\}\) be a set of \(n\) points in Euclidean 3-space \(\mathbb{R}^3\). For \(x_i \in X\), let \(f(x_i)\) denote the number of points in \(X\) at maximal distance from \(x_i\) (``furthest neighbours of \(x_i\)''). In 1988, D. Avis, P. Erdös and J. Pach proved that \[ d(n) : = \max_H \bigl\{ f(x_1) + \cdots + f(x_n) \bigr\} - n^2/4 - 3n/2 \leq c \] for a suitable real constant \(c\). The author sharpens this result by proving the Theorem: Suppose \(n > n_0\) \((n\) sufficiently large), then: (a) if \(n \equiv 0 \bmod 2\) then \(d(n) = 3\); (b) if \(n \equiv 1 \bmod 4\) then \(d(n) = 9/4\); (c) if \(n \equiv 3 \mod 4\) then \(d(n) = 13/4\). Besides he proves that if \(n\) points in \(\mathbb{R}^3\) determine the maximum number of furthest neighbours then they form a ``suspension'', i.e. a finite subset of the union of the polar axis and the equator of a sphere; the optimal arrangement is also given.
scientific article; zbMATH DE number 895989