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b37e936 | 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 | import OAI.Geometry.RCD.Sobolev.LipschitzCompactSubsequence
namespace OAI
/-! # Compactness after subtracting the value at the center -/
open Set Filter Metric
open scoped NNReal Topology
noncomputable section
namespace RegularPointCoordinates
variable {X : Type*} [MetricSpace X] [ProperSpace X]
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
end RegularPointCoordinates
end
end OAI
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