| import OAI.Geometry.CylinderCovering.Convexity.OpenBaseApproximationDefinitions | |
| namespace OAI | |
| noncomputable section | |
| open Set MeasureTheory | |
| namespace RuledApproximation | |
| theorem Cylinder.shadowArea_eq_of_axis_eq {C D : Cylinder} | |
| (haxis : D.axis = C.axis) (S : Set Space) : D.shadowArea S = C.shadowArea S := by | |
| rcases C with ⟨u, hu, B⟩ | |
| rcases D with ⟨v, hv, A⟩ | |
| change v = u at haxis | |
| subst v | |
| rfl | |
| theorem familyRelativeArea_le {n : ℕ} (C : Fin n → Cylinder) (S : Set Space) | |
| {m : ℝ} (hm : 0 < m) (hshadow : ∀ i, m ≤ (C i).shadowArea S) : | |
| familyRelativeArea C S ≤ familyArea C / m := by | |
| unfold familyRelativeArea familyArea | |
| rw [Finset.sum_div] | |
| apply Finset.sum_le_sum | |
| intro i _ | |
| exact div_le_div_of_nonneg_left ENNReal.toReal_nonneg hm (hshadow i) | |
| theorem familyRelativeArea_lt_half {n : ℕ} (C : Fin n → Cylinder) (S : Set Space) | |
| {m : ℝ} (hm : 0 < m) (hshadow : ∀ i, m ≤ (C i).shadowArea S) | |
| (hcost : familyArea C < m / 2) : familyRelativeArea C S < 1 / 2 := by | |
| apply (familyRelativeArea_le C S hm hshadow).trans_lt | |
| rw [div_lt_iff₀ hm] | |
| linarith | |
| end RuledApproximation | |
| end | |
| end OAI | |