Problem 982

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

Source: erdosproblems.com/982

Geometry
Convexity
Distances

Wiki page

Status

Open

No claim settles this problem.

Falsifiable: the site labels the problem FALSIFIABLE, a counterexample being a finite check, and its page was last edited on 19 October 2025. Three refereed lower bounds the site credits settle small cases and have accepted partial claim pages: Moser 1952, Erdős and Fishburn 1994 and Dumitrescu 2006. The proof-claims tab carries one partial claim, a lower bound submitted 2026-07-25 by Scott Duke Kominers, which has no claim page for the reason given under Current assessment; the label was unchanged on 2026-10-06.

BaRo13: Bárány, Imre and Roldán-Pensado, Edgardo, A question from a famous paper of Erdős. Discrete Comput. Geom. (2013), 253-261.
Du06b: Dumitrescu, Adrian, On distinct distances from a vertex of a convex polygon. Discrete Comput. Geom. 36 (2006), 503-509.
Er46b: Erdős, P., On sets of distances of $n$ points. Amer. Math. Monthly (1946), 248-250.
ErFi94: Erdős, Paul and Fishburn, Peter, A postscript on distances in convex $n$ -gons. Discrete Comput. Geom. (1994), 111-117.
Mo52: Moser, Leo, On the different distances determined by $n$ points. Amer. Math. Monthly (1952), 85-91.
NPPZ13: Nivasch, Gabriel and Pach, János and Pinchasi, Rom and Zerbib, Shira, The number of distinct distances from a vertex of a convex polygon. J. Comput. Geom. (2013), 1-12.

Problem 982

Status

Proof claims

Discussion