| import Mathlib |
|
|
| namespace OAI |
|
|
| namespace PlanarL1 |
|
|
| / |
| no vertex in an edge interior, and disjoint interiors for distinct unoriented edges. -/ |
| def IsPlanar {V : Type} (G : SimpleGraph V) : Prop := |
| ∃ (point : V → ℝ × ℝ) (arc : ∀ u v : V, G.Adj u v → C(unitInterval, ℝ × ℝ)), |
| Function.Injective point ∧ |
| (∀ u v h, (arc u v h) 0 = point u ∧ (arc u v h) 1 = point v) ∧ |
| (∀ u v h, Function.Injective (arc u v h)) ∧ |
| (∀ u v h (t : unitInterval) w, 0 < (t : ℝ) → (t : ℝ) < 1 → |
| (arc u v h) t ≠ point w) ∧ |
| (∀ u v h x y k (s t : unitInterval), |
| 0 < (s : ℝ) → (s : ℝ) < 1 → 0 < (t : ℝ) → (t : ℝ) < 1 → |
| (arc u v h) s = (arc x y k) t → |
| (u = x ∧ v = y) ∨ (u = y ∧ v = x)) |
|
|
| / |
| def walkLength {V : Type} {G : SimpleGraph V} (length : V → V → ℝ) |
| {u v : V} : G.Walk u v → ℝ |
| | .nil => 0 |
| | @SimpleGraph.Walk.cons _ _ u w _ _ p => length u w + walkLength length p |
|
|
| / |
| noncomputable def graphDistance {V : Type} (G : SimpleGraph V) |
| (length : V → V → ℝ) (u v : V) : ℝ := |
| sInf (Set.range (fun p : G.Walk u v => walkLength length p)) |
|
|
| / |
| arbitrary symmetric positive edge lengths, realized in a real `L¹` space. -/ |
| def MainStatement : Prop := |
| ∃ C : ℝ, 1 ≤ C ∧ |
| ∀ (n : ℕ) (G : SimpleGraph (Fin n)) (length : Fin n → Fin n → ℝ), |
| G.Connected → IsPlanar G → |
| (∀ u v, length u v = length v u) → |
| (∀ u v, G.Adj u v → 0 < length u v) → |
| ∃ (Ω : Type) (measurable : MeasurableSpace Ω) |
| (μ : @MeasureTheory.Measure Ω measurable) |
| (f : Fin n → @MeasureTheory.Lp Ω ℝ measurable _ 1 μ), |
| ∀ x y, graphDistance G length x y ≤ ‖f x - f y‖ ∧ |
| ‖f x - f y‖ ≤ C * graphDistance G length x y |
|
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| theorem planar_graph_metrics_embed_L1 : MainStatement := by |
| sorry |
|
|
| end PlanarL1 |
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| end OAI |
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|
