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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 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 | import OAI.Geometry.RCD.Metric.PowerPacking
namespace OAI
/-! # Dimensional packing in a pointed target of the actual RCD ball -/
open Set Metric MeasureTheory
noncomputable section
namespace RegularPointCoordinates
variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] [ProperSpace X]
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
end
end OAI
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