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Problem 957 — Erdős Problems

Problem 957

Let $A\subset \mathbb{R}^2$ be a set of size $n$ and let $\{d_1<\ldots<d_k\}$ be the set of distinct distances determined by $A$ . Let $f(d)$ be the number of times the distance $d$ is determined. Is it true that

f(d_1)f(d_k) \leq (\tfrac{9}{8}+o(1))n^2?

Source: erdosproblems.com/957

Geometry
Distances

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Solved·, Proved

An accepted solution exists. The statement is true.

Proved. The site marks the problem proved and credits Dumitrescu's product inequality; see the claim page (Dumitrescu, 2019).

Du19: Dumitrescu, Adrian, A product inequality for extreme distances. Comput. Geom. 85 (2019), 101577, 10. Conference version: 35th International Symposium on Computational Geometry (SoCG 2019), LIPIcs 129, 30:1-30:12.
ErPa90: Erdős, P. and Pach, J., Variations on the theme of repeated distances. Combinatorica 10 (1990), 261-269.

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Status: Solved, Proved
Topics: Geometry
Distances
In the wiki: Wiki page