File size: 2,840 Bytes
695bd70 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 | 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
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