The two largest distances in finite planar sets (Q1916127)

Let \(S\) be a set of \(n\) points in the Euclidean plane \(\mathbb{R}^2\). Let \(n_1\) and \(n_2\) denote the number of times the largest and second largest distances occur among points in \(S\). It is well known that \(n_1 \leq n\). In this paper, a complete set of homogeneous linear inequalities involving \(n\), \(n_1\) and \(n_2\) is given by Theorem: For any set of \(n\) points in \(\mathbb{R}^2\), (0) \(n_1 \leq n\), (i) \(n_2 \leq 3n/2\), (ii) \(n_2 \leq n + 2n_1\), (iii) \(n_1 + n_2 \leq 2n\).