| import OAI.Geometry.RCD.Metric.PowerPacking |
|
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| namespace OAI |
|
|
| /-! # Dimensional packing in a pointed target of the actual RCD ball -/ |
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|
| open Set Metric MeasureTheory |
| noncomputable section |
| namespace RegularPointCoordinates |
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] [ProperSpace X] |
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|
| theorem NoncollapsedRCD.approximate_target_packing {K : ℝ} {n : ℕ} |
| (hX : NoncollapsedRCD K n X) (hn : 2 ≤ n) |
| {Y ι : Type*} [MetricSpace Y] [Fintype ι] {p : X} {q : Y} {ρ d ε : ℝ} |
| (hρ : 0 < ρ) (hd : 0 < d) (hd2 : d ≤ 2) (he : ε ≤ d/6) |
| (hcurv : -1 ≤ K*(4*ρ)^2) {f : X → Y} |
| (hf : IsPointedApproximation p q ρ 1 ε f) (v : ι → Y) |
| (hv : ∀ i, dist (v i) q ≤ 1-ε) (hsep : Pairwise (fun i j => d ≤ dist (v i) (v j))) : |
| (Fintype.card ι:ℝ)*(d/2)^n ≤ Real.cosh 1^(n-1)*8^n := by |
| classical |
| have hex (i : ι) : ∃ x : X, dist x p ≤ ρ ∧ dist (f x) (v i) ≤ ε := by |
| simpa only [one_mul] using hf.2.2 (v i) (hv i) |
| choose x hx hclose using hex |
| have hdist (i j : ι) (hij : i ≠ j) : (d/2)*ρ ≤ dist (x i) (x j) := by |
| have hh := (abs_le.mp (hf.2.1 (x i) (x j) (by simpa using hx i) (by simpa using hx j))).2 |
| have ht := dist_triangle (v i) (f (x i)) (v j) |
| have ht' := dist_triangle (f (x i)) (f (x j)) (v j) |
| have hi := hclose i |
| have hj := hclose j |
| rw [dist_comm (f (x i)) (v i)] at hi |
| have hl := hsep hij |
| apply (le_div_iff₀ hρ).mp |
| linarith |
| have hxin : Function.Injective x := by |
| intro i j h |
| by_contra hij |
| have hh := hdist i j hij |
| rw [h,dist_self] at hh |
| exact (not_le_of_gt (by positivity : 0 < (d/2)*ρ)) hh |
| let S := Finset.univ.image x |
| have hScard : S.card = Fintype.card ι := by |
| dsimp only [S] |
| rw [Finset.card_image_of_injective _ hxin,Finset.card_univ] |
| have hSnear : ∀ y ∈ S, dist y p ≤ ρ := by |
| intro y hy |
| obtain ⟨i,_,rfl⟩ := Finset.mem_image.mp hy |
| exact hx i |
| have hSsep : (S:Set X).Pairwise (fun y z => (d/2)*ρ ≤ dist y z) := by |
| intro y hy z hz hyz |
| obtain ⟨i,_,rfl⟩ := Finset.mem_image.mp hy |
| obtain ⟨j,_,rfl⟩ := Finset.mem_image.mp hz |
| exact hdist i j (fun h => hyz (congrArg x h)) |
| have hb := hX.power_packing_bound hn hρ (by positivity : 0 < (d/2)*ρ) |
| (by nlinarith) hcurv S hSnear hSsep |
| rw [hScard,mul_pow,← mul_assoc] at hb |
| exact (mul_le_mul_iff_left₀ (pow_pos hρ n)).mp hb |
|
|
| end RegularPointCoordinates |
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| end |
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| end OAI |
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