Erdős Problem #660

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 $x_i$?

#660: [Er97e,p.531]

geometry | distances | convex

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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.

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This page was last edited 01 January 2026.

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3 comments on this problem

Likes this problem	Alfaiz
Interested in collaborating	None
Currently working on this problem	None
This problem looks difficult	None
This problem looks tractable	None

Additional thanks to: kiwomuc

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