Problem Statement
Let $x_1,\ldots,x_n\in \mathbb{R}^3$ be the vertices of a convex polyhedron. Are there at least\[(1-o(1))\frac{n}{2}\]many distinct distances between the $x_i$?
Categories:
Geometry Distances Convex
Progress
For the similar problem in $\mathbb{R}^2$ there are always at least $n/2$ distances, as proved by Altman [Al63] (see [93]). In [Er75f] Erdős claims that Altman proved that the vertices determine $\gg n$ many distinct distances, but gives no reference.Source: erdosproblems.com/660 | Last verified: January 16, 2026