| 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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|
