openAI-math-dataset / lean /OAI /Geometry /CylinderCovering /Convexity /OpenBaseApproximationDefinitions.lean
| 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 | |