| import Mathlib |
| import OAI.Geometry.SmoothYau.Spectrum.SphereProjection |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set Filter Manifold Bundle MeasureTheory |
| open scoped Topology ContDiff ENNReal |
| open Set Filter Manifold Bundle |
| open scoped Topology ContDiff |
| open Set Filter Metric |
| open scoped Topology InnerProductSpace |
| open Set Filter Function Metric |
| open scoped Topology |
| open Set Filter Function Metric |
| open scoped Topology |
| open Set Filter Manifold BoxIntegral Metric |
| open scoped Topology ContDiff |
| namespace YauCounterexamples |
|
|
| lemma exists_box_Icc_subset_open {ι : Type*} [Fintype ι] |
| {U : Set (ι → ℝ)} (hU : IsOpen U) (hne : U.Nonempty) : |
| ∃ I : Box ι, Box.Icc I ⊆ U := by |
| obtain ⟨x,hx⟩ := hne |
| obtain ⟨ε,hε,hball⟩ := Metric.isOpen_iff.mp hU x hx |
| let I : Box ι := ⟨fun i => x i-ε/2,fun i => x i+ε/2,fun i => by linarith⟩ |
| refine ⟨I,?_⟩ |
| intro y hy |
| have hd : dist y x ≤ ε/2 := by |
| rw [dist_eq_norm] |
| apply (pi_norm_le_iff_of_nonneg (by positivity)).mpr |
| intro i |
| rw [Real.norm_eq_abs] |
| have h₁ : x i-ε/2 ≤ y i := hy.1 i |
| have h₂ : y i ≤ x i+ε/2 := hy.2 i |
| exact abs_le.mpr ⟨by dsimp; linarith,by dsimp; linarith⟩ |
| exact hball (hd.trans_lt (by linarith)) |
|
|
| theorem exists_spherical_sign_box : |
| ∃ I : Box (Fin 3), ∀ y ∈ |
| (PiLp.continuousLinearEquiv 2 ℝ (fun _ : Fin 3 => ℝ)).symm '' Box.Icc I, |
| (1/20 : ℝ) < sphericalRadius sourceAxisOne sourceAxisTwo |
| ((chartAt (Euclidean 3) sourcePole).symm y) ∧ |
| sphericalRadius sourceAxisOne sourceAxisTwo |
| ((chartAt (Euclidean 3) sourcePole).symm y) < 1/8 := by |
| let e := (PiLp.continuousLinearEquiv 2 ℝ (fun _ : Fin 3 => ℝ)).symm |
| let f : (Fin 3 → ℝ) → ℝ := fun x => |
| sphericalRadius sourceAxisOne sourceAxisTwo ((chartAt (Euclidean 3) sourcePole).symm (e x)) |
| have hf : Continuous f := (sphericalRadius_continuous _ _).comp |
| ((sphere_chart_symm_continuous sourcePole).comp e.continuous) |
| have hsrc : sourceSignPoint ∈ (chartAt (Euclidean 3) sourcePole).source := by |
| apply sphericalRadius_sublevel_in_source sourcePole sourceAxisOne sourceAxisTwo |
| sphericalRadius_neg_sourcePole (R := 1/8) (by norm_num) |
| change sphericalRadius sourceAxisOne sourceAxisTwo sourceSignPoint ≤ 1/8 |
| rw [sphericalRadius_sourceSignPoint] |
| norm_num |
| have hn : (f ⁻¹' Set.Ioo (1/20 : ℝ) (1/8)).Nonempty := by |
| refine ⟨e.symm ((chartAt (Euclidean 3) sourcePole) sourceSignPoint),?_⟩ |
| change 1/20 < f _ ∧ f _ < 1/8 |
| dsimp [f] |
| rw [e.apply_symm_apply,(chartAt (Euclidean 3) sourcePole).left_inv hsrc, |
| sphericalRadius_sourceSignPoint] |
| norm_num |
| obtain ⟨I,hI⟩ := exists_box_Icc_subset_open (isOpen_Ioo.preimage hf) hn |
| refine ⟨I,?_⟩ |
| rintro y ⟨x,hx,rfl⟩ |
| exact hI hx |
| |
| end YauCounterexamples |
| |
| end |
| |
| end OAI |
| |
