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

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

1996-05-10

arxiv.org/abs/math/9605221v1

Mathematics

Metric Geometry

Scientific paper

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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