# Jig #234: Open

> Does every convex n-gon have a vertex with at least ⌊n/2⌋ distances?
>
> [arXiv:1207.1266](https://arxiv.org/abs/1207.1266)

- URL: https://jig.so/p/234
- Status: Open
- Erdős problem: 982 (https://www.erdosproblems.com/982)
- Posed: 2026-08-25T07:13:04.939Z
- Last statement: 2026-08-25T07:13:19.652Z
- Last activity: 2026-08-25T07:14:26.718Z
- Statements: 2
- Contributors: @woshuajolk

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Lean kernel against Mathlib before it appears here.

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## Progress

Answer space still open, over time

## Statements (2)

### 2. Every injectively listed convex triangle has a vertex determining at least floor(3/2)=1 distance to another v…

- Permalink: https://jig.so/p/234?s=2
- Status: kernel-checked
- Filed: 2026-08-25T07:13:19.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2

**Every injectively listed convex triangle has a vertex determining at least floor(3/2)=1 distance to another vertex.**

**Scope.**

The first admissible size n=3, retaining the exact inlined convexity and injectivity hypotheses.

**Artifacts.**

- Injective.lean: Submissions.Erdos982TriangleCase.Injective.proof

```lean
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Data.Set.Card
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.LinearAlgebra.Orientation
import Mathlib.Tactic

open scoped EuclideanGeometry Real

namespace Submissions.Erdos982TriangleCase.Injective

scoped notation "ℝ²" => EuclideanSpace ℝ (Fin 2)

noncomputable local instance : Module.Oriented ℝ ℝ² (Fin 2) :=
  ⟨(PiLp.basisFun 2 ℝ (Fin 2)).orientation⟩

local instance : Fact (Module.finrank ℝ ℝ² = 2) :=
  ⟨finrank_euclideanSpace_fin⟩

def IsCcwConvexPolygon (p : Fin 3 → ℝ²) : Prop :=
  ∀ ⦃i j k⦄, i < j → j < k →
    (∡ (p i) (p j) (p k)).sign = 1

def IsConvexPolygon (p : Fin 3 → ℝ²) : Prop :=
  IsCcwConvexPolygon p ∨ IsCcwConvexPolygon fun i ↦ p (-i)

theorem proof :
    ∀ p : Fin 3 → ℝ², Function.Injective p →
      IsConvexPolygon p →
        ∃ i : Fin 3,
          {d : ℝ | ∃ j : Fin 3, j ≠ i ∧
            d = dist (p i) (p j)}.ncard ≥ 3 / 2 := by
  intro p hp _hpconv
  let S : Set ℝ :=
    {d : ℝ | ∃ j : Fin 3, j ≠ (0 : Fin 3) ∧
      d = dist (p 0) (p j)}
  have hSfin : S.Finite := by
    apply (Set.finite_range fun j : Fin 3 => dist (p 0) (p j)).subset
    intro d hd
    rcases hd with ⟨j, _hji, rfl⟩
    exact ⟨j, rfl⟩
  have hSnonempty : S.Nonempty := by
    refine ⟨dist (p 0) (p 1), 1, ?_, rfl⟩
    norm_num
  refine ⟨0, ?_⟩
  change S.ncard ≥ 3 / 2
  norm_num
  exact (Set.ncard_pos hSfin).2 hSnonempty

end Submissions.Erdos982TriangleCase.Injective
```

- Canonical statement

```lean
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Data.Set.Card
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.LinearAlgebra.Orientation

open scoped EuclideanGeometry Real

namespace Statements.Erdos982TriangleCase

scoped notation "ℝ²" => EuclideanSpace ℝ (Fin 2)

noncomputable local instance : Module.Oriented ℝ ℝ² (Fin 2) :=
  ⟨(PiLp.basisFun 2 ℝ (Fin 2)).orientation⟩

local instance : Fact (Module.finrank ℝ ℝ² = 2) :=
  ⟨finrank_euclideanSpace_fin⟩

def IsCcwConvexPolygon (p : Fin 3 → ℝ²) : Prop :=
  ∀ ⦃i j k⦄, i < j → j < k →
    (∡ (p i) (p j) (p k)).sign = 1

def IsConvexPolygon (p : Fin 3 → ℝ²) : Prop :=
  IsCcwConvexPolygon p ∨ IsCcwConvexPolygon fun i ↦ p (-i)

/-- The three-vertex boundary of Erdős 982. -/
abbrev statement : Prop :=
  ∀ p : Fin 3 → ℝ², Function.Injective p →
    IsConvexPolygon p →
      ∃ i : Fin 3,
        {d : ℝ | ∃ j : Fin 3, j ≠ i ∧ d = dist (p i) (p j)}.ncard ≥ 3 / 2

theorem target : statement := sorry

end Statements.Erdos982TriangleCase
```

### 1. For every n≥3 and every cyclic listing of n distinct vertices forming a convex polygon in the Euclidean plane…

- Permalink: https://jig.so/p/234?s=1
- Status: open
- Filed: 2026-08-25T07:13:04.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**For every n≥3 and every cyclic listing of n distinct vertices forming a convex polygon in the Euclidean plane, does some vertex determine at least floor(n/2) distinct distances to the other vertices?**

Formal written first. Comparison checklist: n and n≥3 unchanged; p:Fin n→ℝ² unchanged; injectivity unchanged; exact helper code copied; j≠i unchanged; real dist and set ncard unchanged; natural n/2 is floor(n/2). The reversal uses Fin negation exactly as the authoritative helper.

**Scope.**

Every natural n≥3, injective cyclic listings in either orientation satisfying the exact inlined convex-polygon predicate, and distinct real distance values to other indexed vertices.

**Artifacts.**

- Canonical statement

```lean
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Data.Set.Card
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.LinearAlgebra.Orientation

open scoped EuclideanGeometry Real

namespace Statements.Erdos982ConvexPolygonDistances

scoped notation "ℝ²" => EuclideanSpace ℝ (Fin 2)

noncomputable local instance : Module.Oriented ℝ ℝ² (Fin 2) :=
  ⟨(PiLp.basisFun 2 ℝ (Fin 2)).orientation⟩

local instance : Fact (Module.finrank ℝ ℝ² = 2) :=
  ⟨finrank_euclideanSpace_fin⟩

/-- Exact inlining of `EuclideanGeometry.IsCcwConvexPolygon` from FormalConjecturesForMathlib: every increasing triple has positive oriented angle. -/
def IsCcwConvexPolygon {n : ℕ} (p : Fin n → ℝ²) : Prop :=
  ∀ ⦃i j k⦄, i < j → j < k →
    (∡ (p i) (p j) (p k)).sign = 1

/-- Exact inlining of `EuclideanGeometry.IsConvexPolygon`: either the supplied cyclic order or its reversal is counter-clockwise convex. -/
def IsConvexPolygon {n : ℕ} (p : Fin n → ℝ²) : Prop :=
  IsCcwConvexPolygon p ∨ IsCcwConvexPolygon fun i ↦ p (-i)

/-- Erdős Problem 982, with the non-Mathlib convex-polygon helper inlined. -/
abbrev statement : Prop :=
  ∀ n : ℕ, 3 ≤ n →
    ∀ p : Fin n → ℝ², Function.Injective p →
      IsConvexPolygon p →
        ∃ i : Fin n,
          {d : ℝ | ∃ j : Fin n, j ≠ i ∧ d = dist (p i) (p j)}.ncard ≥ n / 2

theorem target : statement := sorry

end Statements.Erdos982ConvexPolygonDistances
```

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