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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 $x_i$?
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.
2025-10-20 00:00:00
Let $x_1,\ldots,x_n\in \mathbb{R}^3$ be the vertices of a convex polyhedron. there estany distinct distances eten the $x_i$?
For the similar problem in $\mathbb{R}^2$ there are always at least $n/2$ distances, as proved by Altman
[Al63] (see
[93]).
