Problem Statement
Let $x_1,\ldots,x_n\in\mathbb{R}^2$ determine the set of distances $\{u_1,\ldots,u_t\}$. Suppose $u_i$ appears as the distance between $f(u_i)$ many pairs of points. Then for all $\epsilon>0$\[\sum_i f(u_i)^2 \ll_\epsilon n^{3+\epsilon}.\]
Categories:
Geometry Convex Distances
Progress
The case when the points determine a convex polygon was solved by Fishburn [Al63]. Note it is trivial that $\sum f(u_i)=\binom{n}{2}$. Solved by Guth and Katz [GuKa15] who proved the upper bound\[ \sum_i f(u_i)^2 \ll n^3\log n.\]See also [94].Source: erdosproblems.com/95 | Last verified: January 13, 2026