OPEN
Let $x_1,\ldots,x_n\in \mathbb{R}^3$ be the vertices of a convex polyhedron. Is there some constant $c>0$ such that, for all $n$ sufficiently large, the number of distinct distances determined by the $x_i$ is at most
\[(1-c)\frac{n}{2}?\]
For the similar problem in $\mathbb{R}^2$ there are always at least $n/2$ distances, as proved by Altman
[Al63] (see
[93]).
