Erdős Problem #660
ambiguous statement
OPENdatabase updated 2025-08-31
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
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`P` is the set of vertices of a (full-dimensional) convex polyhedron in \(\mathbb{R}^3\): the points are in convex position and they affinely span \(\mathbb{R}^3\) (so the polyhedron is genuinely three-dimensional).
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 \(x_i\)? The \((1 - o(1)) \frac{n}{2}\) lower bound is formalised as: for every \(\varepsilon > 0\), every set of \(n\) vertices of a convex polyhedron with \(n\) sufficiently large determines at least \((1 - \varepsilon) \frac{n}{2}\) distinct distances.
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open · 2 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/660.Attempts
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Published few-distance classifications plus exact witnesses determine \(f_{\rm cvx}(n)\) for every \(4\le n\le20\) (with a jump from 3 to 5), and an elementary regular-pyramid family attains \(\lfloor
PARTIAL