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Erdős #982 · geometry / convex

If nn distinct points in R2\mathbb{R}^2 form a convex polygon then some vertex has at least ⌊n2⌋\lfloor \frac{n}{2}\rfloor different distances to other vertices.

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Problem statement

If nn distinct points in R2\mathbb{R}^2 form a convex polygon then some vertex has at least ⌊n2⌋\lfloor \frac{n}{2}\rfloor different distances to other vertices.

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Background

The regular polygon shows that ⌊n/2⌋\lfloor n/2\rfloor is the best possible here.

This would be implied if there was a vertex such that no three vertices of the polygon are equally distant to it, which was originally also conjectured by Erdős [Er46b], but this is false (see [97]).

Let f(n)f(n) be the maximal number of such distances that are guaranteed. Moser [Mo52] proved that

f(n)≥⌈n3⌉.f(n) \geq \left\lceil\frac{n}{3}\right\rceil.

This was improved by Erdős and Fishburn [ErFi94] to

f(n)≥⌊n3+1⌋,f(n) \geq \left\lfloor \frac{n}{3}+1\right\rfloor,

then

f(n)≥⌈13n−636⌉f(n) \geq \left\lceil \frac{13n-6}{36}\right\rceil

by Dumitrescu [Du06b], and most recently

f(n)≥(1336+122701)n−O(1)f(n) \geq \left(\frac{13}{36}+\frac{1}{22701}\right)n-O(1)

by Nivasch, Pach, Pinchasi, and Zerbib [NPPZ13].

In [Er46b] Erdős makes the even stronger conjecture that on every convex curve there exists a point pp such that every circle with centre pp intersects the curve in at most 22 points. Bárány and Roldán-Pensado [BaRo13] noted that the boundary of any acute triangle is a counterexample.

Bárány and Roldán-Pensado prove that, for any planar convex body, there is a point pp on the boundary such that every circle with centre pp intersects the boundary in at most O(1)O(1) (where the implied constant depends on the convex body). They conjecture that there this can be bounded by an absolute constant - that is, Erdős's conjecture is true if we replace 22 by some larger constant CC.

See also [93].

Progress & source assessment

The source labels this entry FALSIFIABLE in the 2026-09-08 snapshot. Editorial progress notes are reproduced as source commentary, not independently verified results. Prize amounts and Lean badges at the source are not local awards or verification.

References
  1. [1]Thomas Bloom (ed.), Erdős Problems #982 link
  2. [2]Upstream Lean 4 statement for Erdős #982 linkExternal statement reference. Not compiled or verified by this platform; upstream files may contain open proof obligations.
  3. [3]Formal Conjectures — Erdős Problem 982 link

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