Problem Statement
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\[\sum_i f(u_i)^2 \ll n^3.\]
Categories:
Geometry Convex Distances
Progress
In [Er97c] Erdős claims that Fishburn solved this, but gives no reference. Note it is trivial that $\sum f(u_i)=\binom{n}{2}$.Lefmann and Theile [LeTh95] prove a stronger version of this question, that\[\sum_i f(u_i)^2 \ll n^3\]under the weaker assumption that no three points are on a line. A sketch of this proof is given in the comments by serge.
Erdős and Fishburn also make the stronger conjecture that $\sum f(u_i)^2$ is maximal for the regular $n$-gon (for large enough $n$).
In [Er92e] Erdős offered £25 for a resolution of this problem. (This has been converted to $44 using approximate 1992 exchange rates.)
See also [95].
Source: erdosproblems.com/94 | Last verified: January 13, 2026