openAI-math-dataset / lean /OAI /Geometry /Perimeter /Regularity /ClosedFilledLineInvariantOfFrontier.lean
| import OAI.Geometry.Perimeter.Regularity.ExistsMinimizingDensityLimit | |
| namespace OAI | |
| noncomputable section | |
| open Set Filter MeasureTheory | |
| open scoped Topology ENNReal NNReal ContDiff | |
| open scoped Topology ENNReal NNReal | |
| open scoped ENNReal Topology ContDiff Pointwise | |
| open scoped Topology ENNReal ContDiff | |
| open scoped BoundedContinuousFunction | |
| open scoped ENNReal Topology | |
| open scoped ENNReal Topology ContDiff | |
| open scoped Topology ENNReal | |
| namespace RPISingularHarnack | |
| theorem closed_filled_line_invariant_of_frontier {n : ℕ} {F : Set (Euc n)} | |
| (hF : IsClosed F) (c : Euc n) | |
| (hline : ∀ (t : ℝ) (x : Euc n), x ∈ frontier F → x+t • c ∈ frontier F) : | |
| ∀ (t : ℝ) (x : Euc n), x ∈ F → x+t • c ∈ F := by | |
| intro t x hx | |
| by_cases hxfr : x ∈ frontier F | |
| · exact hF.closure_eq ▸ frontier_subset_closure (hline t x hxfr) | |
| have hxint : x ∈ interior F := by | |
| by_contra hxi | |
| apply hxfr | |
| rw [frontier, hF.closure_eq] | |
| exact ⟨hx, hxi⟩ | |
| let L : Set (Euc n) := Set.range (fun s : ℝ ↦ x+s • c) | |
| have hLfr : Disjoint L (frontier F) := by | |
| apply Set.disjoint_left.mpr | |
| rintro y ⟨s, rfl⟩ hy | |
| apply hxfr | |
| have hh := hline (-s) (x+s • c) hy | |
| convert hh using 1 | |
| module | |
| have hLsub : L ⊆ interior F ∪ Fᶜ := by | |
| intro y hy | |
| by_cases hyF : y ∈ F | |
| · apply Or.inl | |
| by_contra hyi | |
| exact (Set.disjoint_left.mp hLfr hy) (by rw [frontier, hF.closure_eq]; exact ⟨hyF, hyi⟩) | |
| · exact Or.inr hyF | |
| have hLc : IsPreconnected L := by | |
| exact isPreconnected_range (by fun_prop : Continuous (fun s : ℝ ↦ x+s • c)) | |
| have hLInt : L ⊆ interior F := hLc.subset_left_of_subset_union isOpen_interior hF.isOpen_compl | |
| (Set.disjoint_left.mpr fun y hy hyc ↦ hyc (interior_subset hy)) hLsub | |
| ⟨x, ⟨0, by simp⟩, hxint⟩ | |
| exact interior_subset (hLInt ⟨t, rfl⟩) | |
| theorem minimizing_cone_tangent_filled_line_invariant {n : ℕ} (hn : 1 < n) | |
| {E F : Set (Euc n)} (hE : IsLocallyPerimeterMinimizing E) (hcone : IsCone E) | |
| (hF : IsLocallyPerimeterMinimizing F) (hFc : IsClosed F) | |
| (hFfront : frontier F = perimeterSupport F) (c : Euc n) | |
| (r : ℕ → ℝ) (hr : ∀ i, 0 < r i) (htr : Tendsto r atTop (𝓝 0)) | |
| (hconv : ∀ (x : Euc n) (R : ℝ), 0 < R → | |
| Tendsto (fun i ↦ volume.real (symmDiff (blownUpSet E c (r i)) F ∩ Metric.closedBall x R)) | |
| atTop (𝓝 0)) : | |
| ∀ (t : ℝ) (x : Euc n), x ∈ F → x + t • c ∈ F := by | |
| apply closed_filled_line_invariant_of_frontier hFc c | |
| intro t x hx | |
| rw [hFfront] at hx ⊢ | |
| exact minimizing_cone_tangent_support_line_invariant hn hE hcone hF c r hr htr hconv t x hx | |
| end RPISingularHarnack | |
| end | |
| end OAI | |