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Convex and discrete geometry
Erdős problem 982
If n distinct points in R2 form a convex polygon then some vertex has at least
⌊2n⌋
different distances to other vertices.
$2,697 bounty·1 piece from 1 person·last one 18 days ago
week of 27 Sept 2025 · nothing publishedweek of 4 Oct 2025 · nothing publishedweek of 11 Oct 2025 · nothing publishedweek of 18 Oct 2025 · nothing publishedweek of 25 Oct 2025 · nothing publishedweek of 1 Nov 2025 · nothing published
Solve it
A complete proof or refutation in Lean. The first one the validator accepts takes the whole bounty.
∀ (n : ℕ),
3 ≤ n →
∀ (p : Fin n → EuclideanSpace ℝ (Fin 2)),
Function.Injective p →
EuclideanGeometry.IsConvexPolygon p → ∃ i, {d | ∃ j, j ≠ i ∧ d = dist (p i) (p j)}.ncard ≥ n / 2
27 Sept 2025a year of work on this problem19 Sept 2026
erdosproblems.com/982
Something wrong with this formalization?
A statement that does not faithfully capture the original conjecture is the one real risk here, so we would rather hear about it early - before someone spends weeks on it.
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