Problem 94

Suppose $n$ points in $\mathbb{R}^2$ determine a convex polygon and the set of distances between them is $\{u_1,\ldots,u_t\}$ . Suppose $u_i$ appears as the distance between $f(u_i)$ many pairs of points. Then

\sum_i f(u_i)^2 \ll n^3.

Source: erdosproblems.com/94

Geometry
Convexity
Distances

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SolvedProved

An accepted solution exists. The statement is true.

Proved. The site's export of 2026-09-04 records the label "PROVED (LEAN)". The proof is Lefmann and Thiele's theorem for point sets with no three on a line, recorded on its claim page (Lefmann and Thiele, 1995); the Lean qualification refers to third-party developments that this corpus has neither built nor audited (see Formalization). Erdős's 1997 attribution of a proof to Fishburn, given without a reference, has no manuscript to page.

Er92e: Erdős, Pál, Some Unsolved problems in Geometry, Number Theory and Combinatorics. Eureka (1992), 44-48.
Er97c: Erdős, Paul, Some of my favorite problems and results. The mathematics of Paul Erdős, I, Algorithms Combin. 13, Springer (1997), 47--67; printed p. 65: "I conjectured and Fishburn proved that $\sum_is_i^2<cn^3$ " for the distance multiplicities $s_i$ of a convex polygon, stated without proof or reference. Library home: Erdős 1997; paged at Erdős 1997, Conjecture (p. 65).
LeTh95: Lefmann, Hanno and Thiele, Torsten, Point sets with distinct distances. Combinatorica (1995), 379-408.

Problem 94

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