| import OAI.Geometry.RCD.Sobolev.LipschitzCompactSubsequence |
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| namespace OAI |
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| /-! # Compactness after subtracting the value at the center -/ |
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|
| open Set Filter Metric |
| open scoped NNReal Topology |
| noncomputable section |
| namespace RegularPointCoordinates |
| variable {X : Type*} [MetricSpace X] [ProperSpace X] |
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| theorem centered_ball_subsequence (z : X) {r : ℝ} (hr : 0<r) |
| (f : ℕ → X → ℝ) (L : ℝ≥0) |
| (hl : ∀ k, LipschitzOnWith L (f k) (ball z (r+1))) : |
| ∃ (g : closedBall z r → ℝ) (a : ℕ → ℕ), StrictMono a ∧ LipschitzWith L g ∧ |
| g ⟨z,mem_closedBall_self hr.le⟩=0 ∧ |
| TendstoUniformly (fun k (x : closedBall z r) => f (a k) x-f (a k) z) g atTop := by |
| let Y := closedBall z r |
| let : CompactSpace Y := isCompact_iff_compactSpace.mp (isCompact_closedBall z r) |
| let U := fun k (x : Y) => f k x-f k z |
| let z0 : Y := ⟨z,mem_closedBall_self hr.le⟩ |
| have hsub : closedBall z r ⊆ ball z (r+1) := closedBall_subset_ball (by linarith) |
| have hL (k : ℕ) : LipschitzWith L (U k) := by |
| have hres : LipschitzWith L (fun x : Y => f k x) := by |
| apply LipschitzWith.of_dist_le_mul |
| intro x y |
| exact (hl k).dist_le_mul x (hsub x.property) y (hsub y.property) |
| simpa only [U,add_zero] using hres.sub (LipschitzWith.const (f k z)) |
| have hb (k : ℕ) (x : Y) : |U k x|≤(L:ℝ)*r := by |
| have hh := (hL k).dist_le_mul x z0 |
| have hx : dist x.val z≤r := x.property |
| simp only [U,z0,Real.dist_eq,sub_self,sub_zero,Subtype.dist_eq] at hh |
| exact hh.trans (mul_le_mul_of_nonneg_left hx L.coe_nonneg) |
| obtain ⟨g,a,ha,hg,_hgb,hlim⟩ := exists_uniform_lipschitz_subsequence U L ((L:ℝ)*r) hL hb |
| refine ⟨g,a,ha,hg,?_,hlim⟩ |
| have hz : Tendsto (fun k => U (a k) z0) atTop (𝓝 0) := by |
| simpa only [U,z0,sub_self] using (tendsto_const_nhds : |
| Tendsto (fun _ : ℕ => (0:ℝ)) atTop (𝓝 0)) |
| exact tendsto_nhds_unique (hlim.tendsto_at z0) hz |
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| end RegularPointCoordinates |
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| end |
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| end OAI |
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