heilbronn_k_point_conjecture

⚠️ Retired — specification defect

The Lean statement below does not encode the problem shown on this page, so its Disproved status carries no information about that problem. Do not import this node or use it as a dependency.

Heilbronn's k-point problem: For any n points in the unit square, there are k points with convex hull area ≥ c/n^{2/(k-1)}. Best known bounds have significant gaps between upper and lower bounds for k ≥ 4.

Why this node was retired

The posted statement is

import Mathlib

theorem heilbronn_k_point_conjecture (k : ℕ) (hk : 4 ≤ k) :
    ∀ eps : ℝ, 0 < eps →
    ∃ C : ℝ, 0 < C ∧
    ∀ (n : ℕ) (pts : Fin n → ℝ × ℝ),
      ∃ (S : Finset (Fin n)) (_ : S.card = k),
        ∀ i ∈ S, ∀ j ∈ S, ∀ l ∈ S, i ≠ j → j ≠ l → i ≠ l →
          |(pts i).1 * ((pts j).2 - (pts l).2) +
           (pts j).1 * ((pts l).2 - (pts i).2) +
           (pts l).1 * ((pts i).2 - (pts j).2)| / 2 ≥ C * n ^ (-2 + eps) := by
  sorry

The conclusion requires a k-element subset even when n=0; universal positive area claims also fail for repeated or collinear points.

The recorded counterexample refutes the statement as encoded. It says nothing about the problem shown above, which is a different proposition.

Proposed corrected statement

Define the appropriate Heilbronn extremal quantity by optimizing over n-point configurations in a bounded Euclidean region and taking the minimum area over the required distinct tuples. Add n≥k and state the intended asymptotic upper/lower estimate for that extremum; a universal positive lower bound for every configuration is not correct.

Diagnosis and correction from the public Prove2Me statement audit (wamlat/prove2me-errors). The correction is natural-language mathematics and is not Lean-verified — it is a specification for a corrected node, not a drop-in replacement. No corrected replacement node exists yet.