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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 | import OAI.Geometry.RuledCovering.Model
namespace OAI
noncomputable section
open Set MeasureTheory
namespace RuledApproximation
def Cylinder.shadowArea (C : Cylinder) (S : Set Space) : ℝ :=
volume.real (((Submodule.span ℝ ({C.axis} : Set Space))ᗮ).orthogonalProjectionOnto '' S)
def familyRelativeArea {n : ℕ} (C : Fin n → Cylinder) (S : Set Space) : ℝ :=
∑ i, (C i).area / (C i).shadowArea S
def OpenApproximationStatement : Prop :=
∀ (n : ℕ) (C : Fin n → Cylinder) (S : Set Space) (m : ℝ),
(∀ i, IsCompact (C i).base) → S ⊆ familySet C → 0 < m →
(∀ i, m ≤ (C i).shadowArea S) → familyArea C < m / 2 →
∃ D : Fin n → Cylinder,
(∀ i, (D i).axis = (C i).axis) ∧
(∀ i, (D i).openBounded) ∧ S ⊆ familySet D ∧
familyArea D < m / 2 ∧ familyRelativeArea D S < 1 / 2
end RuledApproximation
end
end OAI
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