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Uniformly distributed distances: A geometric application of Jansen's inequality

János Pach·Joel Spencer
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Abstract

Let $d_1\leq d_2\leq\ldots\leq d_{n\choose 2}$ denote the distances determined by $n$ points in the plane. It is shown that $\min\sum_i (d_{i+1}-d_i)^2=O(n^{-6/7})$, where the minimum is taken over all point sets with minimal distance $d_1 \geq 1$. This bound is asymptotically tight.

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János Pach, Joel Spencer. 1996-05-10. Uniformly distributed distances: A geometric application of Jansen's inequality. https://arxiv.org/abs/math/9605221

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