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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 | import OAI.Geometry.CylinderCovering.Convexity.OpenBaseComparison
import OAI.Geometry.CylinderCovering.Convexity.OpenBaseThickening
namespace OAI
noncomputable section
open Set MeasureTheory
namespace RuledApproximation
theorem Cylinder.enlarge_open_compact {C : Cylinder} (hC : IsCompact C.base)
{eps : ℝ} (heps : 0 < eps) :
∃ D : Cylinder, D.axis = C.axis ∧ D.openBounded ∧
C.carrier ⊆ D.carrier ∧ D.area < C.area + eps := by
obtain ⟨O, hCO, hOo, hOb, hOm, hOI⟩ :=
CylinderCovering.OpenBaseThickening.exists_open_enlargement volume hC heps
refine ⟨⟨C.axis, C.unit_axis, O⟩, rfl, ⟨hOo, hOb, hOm⟩, ?_, hOI⟩
rintro x ⟨b, hb, t, hx⟩
exact ⟨b, hCO hb, t, hx⟩
theorem family_enlarge_open_compact {n : ℕ} (C : Fin n → Cylinder)
(hC : ∀ i, IsCompact (C i).base) {eps : ℝ} (heps : 0 < eps) :
∃ D : Fin n → Cylinder, (∀ i, (D i).axis = (C i).axis) ∧
(∀ i, (D i).openBounded) ∧ familySet C ⊆ familySet D ∧
familyArea D ≤ familyArea C + eps := by
classical
have hd : 0 < eps / (n + 1 : ℝ) := div_pos heps (by positivity)
choose D haxis hD hCD harea using fun i => Cylinder.enlarge_open_compact (hC i) hd
refine ⟨D, haxis, hD, ?_, ?_⟩
· intro x hx
obtain ⟨i, hi⟩ := mem_iUnion.mp hx
exact mem_iUnion.mpr ⟨i, hCD i hi⟩
· change (∑ i, (D i).area) ≤ familyArea C + eps
calc
_ ≤ ∑ i, ((C i).area + eps / (n + 1 : ℝ)) :=
Finset.sum_le_sum (fun i _ => (harea i).le)
_ = familyArea C + n * (eps / (n + 1 : ℝ)) := by
simp [familyArea, Finset.sum_add_distrib]
_ ≤ familyArea C + eps := by
have hn : 0 < (n + 1 : ℝ) := by positivity
have hne : (n : ℝ) * (eps / (n + 1 : ℝ)) ≤ eps := by
rw [← mul_div_assoc, div_le_iff₀ hn]
nlinarith
linarith
theorem family_enlarge_open_two_budgets {n : ℕ} (C : Fin n → Cylinder)
(S : Set Space) (hC : ∀ i, IsCompact (C i).base) {A B : ℝ}
(hA : familyArea C < A) (hB : familyRelativeArea C S < B) :
∃ D : Fin n → Cylinder, (∀ i, (D i).axis = (C i).axis) ∧
(∀ i, (D i).openBounded) ∧ familySet C ⊆ familySet D ∧
familyArea D < A ∧ familyRelativeArea D S < B := by
classical
have hw : (∑ i, ((C i).shadowArea S)⁻¹ * volume.real (C i).base) < B := by
simpa only [familyRelativeArea, Cylinder.area, div_eq_mul_inv, mul_comm] using hB
obtain ⟨U, hU, harea, hweighted⟩ :=
CylinderCovering.OpenBaseThickening.exists_open_family_two_budgets
(fun i => (volume : Measure ((Submodule.span ℝ ({(C i).axis} : Set Space))ᗮ)))
(fun i => (C i).base) hC (fun i => ((C i).shadowArea S)⁻¹) hA hw
let D : Fin n → Cylinder := fun i => ⟨(C i).axis, (C i).unit_axis, U i⟩
refine ⟨D, fun _ => rfl, fun i => ⟨(hU i).2.1, (hU i).2.2.1, (hU i).2.2.2⟩,
?_, harea, ?_⟩
· intro x hx
obtain ⟨i, b, hb, t, ht⟩ := mem_iUnion.mp hx
exact mem_iUnion.mpr ⟨i, b, (hU i).1 hb, t, ht⟩
· simpa only [familyRelativeArea, Cylinder.area, Cylinder.shadowArea,
D, div_eq_mul_inv, mul_comm] using hweighted
theorem openApproximation : OpenApproximationStatement := by
intro n C S m hC hcov hm hshadow hcost
have hrelative := familyRelativeArea_lt_half C S hm hshadow hcost
obtain ⟨D, haxis, hopen, hCD, harea, hrel⟩ :=
family_enlarge_open_two_budgets C S hC hcost hrelative
exact ⟨D, haxis, hopen, hcov.trans hCD, harea, hrel⟩
end RuledApproximation
end
end OAI
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