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785c727 | 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 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 | import OAI.Geometry.RuledCovering.Main
import OAI.Geometry.CylinderCovering.Ruled.Similarity
namespace OAI
open Set MeasureTheory Metric
open scoped Topology
noncomputable section
namespace RuledApproximation
theorem compact_coordinates_parallelogram_cover {K : Set Coordinates}
(hK : IsCompact K) {ε : ℝ} (hε : 0 < ε) :
∃ n : ℕ, ∃ P : Fin n → Set Coordinates,
(∀ i, IsParallelogram (P i)) ∧ K ⊆ ⋃ i, P i ∧
(∑ i, volume.real (P i)) < volume.real K + ε := by
classical
have hmodel : LatticeModel (0 : Coordinates →L[ℝ] Coordinates) 1 (1/2) := by
simpa using nilpotent_lattice_model 0
obtain ⟨m, hm, S, hS, _, hcover, hcost⟩ := local_finite_certificate
hK isOpen_univ (subset_univ _) (fun _ => 0) 0
(show (0:ℝ) ≤ 1 by norm_num) (show (0:ℝ) ≤ 0 by norm_num) hmodel
(by intros; simp) (continuous_const : Continuous (fun _ : Coordinates => (1:ℝ)))
(fun _ => zero_le_one) hε
let : Fintype S := hS.fintype
let a : ℝ := (1/2) / m
have hmR : (0:ℝ) < m := by exact_mod_cast hm
have ha : 0 < a := div_pos (by norm_num) hmR
let e := Fintype.equivFin S
let P : Fin (Fintype.card S) → Set Coordinates :=
fun i => closedBall ((e.symm i : S) : Coordinates) a
refine ⟨Fintype.card S, P, fun i => coordinates_closedBall_parallelogram _ ha, ?_, ?_⟩
· intro p hp
obtain ⟨b, hb, hdist⟩ := hcover p hp 0 (by norm_num)
refine mem_iUnion.mpr ⟨e ⟨b, hb⟩, ?_⟩
change p ∈ closedBall ((e.symm (e ⟨b, hb⟩) : S) : Coordinates) a
simp only [Equiv.symm_apply_apply]
rw [mem_closedBall, dist_eq_norm]
simpa [a] using hdist
· have harea (b : S) : volume.real (closedBall (b : Coordinates) a) = 1 / (m:ℝ)^2 := by
rw [coordinates_closedBall_volume _ ha.le]
dsimp [a]
ring
have hsum : (∑ i, volume.real (P i)) = (∑' b : S, (1:ℝ)) / (m:ℝ)^2 := by
change (∑ i, volume.real (closedBall ((e.symm i : S) : Coordinates) a)) = _
rw [e.symm.sum_comp (fun b : S => volume.real (closedBall (b : Coordinates) a))]
simp only [harea, tsum_fintype, Finset.sum_div]
rw [hsum]
simpa only [integral_const, measureReal_restrict_apply_univ, smul_eq_mul, mul_one] using hcost
theorem compact_parallelogram_cover {E : Type*} [NormedAddCommGroup E]
[InnerProductSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
(hdim : Module.finrank ℝ E = 2) {K : Set E} (hK : IsCompact K)
{ε : ℝ} (hε : 0 < ε) :
∃ n : ℕ, ∃ P : Fin n → Set E,
(∀ i, IsParallelogram (P i)) ∧ K ⊆ ⋃ i, P i ∧
(∑ i, volume.real (P i)) < volume.real K + ε := by
let e : E ≃ₗᵢ[ℝ] Plane := ((stdOrthonormalBasis ℝ E).reindex (finCongr hdim)).repr
let M : Coordinates ≃L[ℝ] E :=
(EuclideanSpace.equiv (Fin 2) ℝ).symm.trans e.symm.toContinuousLinearEquiv
have hmp : MeasurePreserving M volume volume :=
e.symm.measurePreserving.comp (PiLp.volume_preserving_toLp (Fin 2))
have harea (T : Set Coordinates) : volume.real (M '' T) = volume.real T := by
have h := hmp.setIntegral_image_emb M.toHomeomorph.measurableEmbedding
(fun _ => (1:ℝ)) T
simpa only [integral_const, measureReal_def, Measure.restrict_apply_univ,
smul_eq_mul, mul_one] using h
let Kc := M.symm '' K
have hKc : IsCompact Kc := hK.image M.symm.continuous
have himage : M '' Kc = K := by simp [Kc]
have hvolume : volume.real Kc = volume.real K := by
rw [← harea Kc, himage]
obtain ⟨n, P, hP, hcover, hcost⟩ := compact_coordinates_parallelogram_cover hKc hε
refine ⟨n, fun i => M '' P i, fun i => (hP i).linear_image M.toLinearMap M.injective,
?_, ?_⟩
· intro x hx
have hx' : M.symm x ∈ Kc := ⟨x, hx, rfl⟩
obtain ⟨i, hi⟩ := mem_iUnion.mp (hcover hx')
exact mem_iUnion.mpr ⟨i, M.symm x, hi, M.apply_symm_apply x⟩
· simp only [harea]
rwa [hvolume] at hcost
theorem Cylinder.compact_parallelogram_cover (C : Cylinder) (hC : IsCompact C.base)
{ε : ℝ} (hε : 0 < ε) :
HasFiniteCover Cylinder.parallelogram C.carrier (C.area + ε) := by
obtain ⟨n, P, hP, hcover, hcost⟩ := RuledApproximation.compact_parallelogram_cover
(similarityPlane_finrank C.axis C.unit_axis) hC hε
let D : Fin n → Cylinder := fun i => ⟨C.axis, C.unit_axis, P i⟩
refine ⟨n, D, hP, ?_, hcost.le⟩
rintro x ⟨b, hb, s, rfl⟩
obtain ⟨i, hi⟩ := mem_iUnion.mp (hcover hb)
exact mem_iUnion.mpr ⟨i, b, hi, s, rfl⟩
lemma HasFiniteCover.finite_iUnion {ι : Type*} [Fintype ι]
{shape : Cylinder → Prop} (S : ι → Set Space) (A : ι → ℝ)
(h : ∀ i, HasFiniteCover shape (S i) (A i)) :
HasFiniteCover shape (⋃ i, S i) (∑ i, A i) := by
classical
have aux (s : Finset ι) : HasFiniteCover shape (⋃ i ∈ s, S i) (∑ i ∈ s, A i) := by
induction s using Finset.induction_on with
| empty => simpa using HasFiniteCover.empty shape (show (0:ℝ) ≤ 0 by norm_num)
| @insert i s hi ih =>
have heq : (⋃ j ∈ insert i s, S j) = S i ∪ ⋃ j ∈ s, S j := by
ext x
simp
rw [heq, Finset.sum_insert hi]
exact (h i).union ih
simpa using aux Finset.univ
theorem compact_family_parallelogram_cover {n : ℕ} (C : Fin n → Cylinder)
(hC : ∀ i, IsCompact (C i).base) {ε : ℝ} (hε : 0 < ε) :
HasFiniteCover Cylinder.parallelogram (familySet C) (familyArea C + ε) := by
have hδ : 0 < ε / (n + 1 : ℝ) := div_pos hε (by positivity)
have hc := HasFiniteCover.finite_iUnion (fun i => (C i).carrier)
(fun i => (C i).area + ε / (n + 1 : ℝ))
(fun i => (C i).compact_parallelogram_cover (hC i) hδ)
apply hc.mono Subset.rfl
simp only [Finset.sum_add_distrib, Finset.sum_const, Finset.card_univ,
Fintype.card_fin, nsmul_eq_mul, familyArea]
have hn : 0 < (n + 1 : ℝ) := by positivity
have hmul : (n : ℝ) * (ε / (n + 1 : ℝ)) ≤ ε := by
rw [← mul_div_assoc, div_le_iff₀ hn]
nlinarith only [hε]
linarith only [hmul]
end RuledApproximation
end
end OAI
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