FormalConjectures.ErdosProblems.«982»

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import FormalConjecturesUtil

Erdős Problem 982

Reference: erdosproblems.com/982


open EuclideanGeometry


namespace Erdos982

If $n$ distinct points in $\mathbb{R}^2$ form a convex polygon then some vertex has at least $\lfloor\frac{n}{2}\rfloor$ different distances to other vertices.

@[category research open, AMS 52]
theorem declaration uses 'sorry'erdos_982 (n : ℕ) (hn : 3 ≤ n) (p : Fin n → ℝ²) (hp : Function.Injective p)
    (hp' : EuclideanGeometry.IsConvexPolygon p) :
    ∃ (i : Fin n), { d : ℝ | ∃ j : Fin n, j ≠ i ∧ d = dist (p i) (p j) }.ncard ≥ n / 2 := byn:ℕhn:3 ≤ np:Fin n → ℝ²hp:Function.Injective php':IsConvexPolygon p⊢ ∃ i, {d | ∃ j, j ≠ i ∧ d = dist (p i) (p j)}.ncard ≥ n / 2
  sorryAll goals completed! 🐙


end Erdos982