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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 | 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
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