| import Mathlib |
| import OAI.Geometry.IntegralFillings.Optimality.BallCoarea |
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| namespace OAI |
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| section |
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| open Set Filter MeasureTheory |
| open scoped Topology ENNReal NNReal |
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| namespace SharpIntegralFillings.Optimality |
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| lemma exists_le_of_integral_le_ae {α : Type*} [MeasurableSpace α] {μ : Measure α} |
| (hμ : μ ≠ 0) {f g : α → ℝ} (hf : Integrable f μ) (hg : Integrable g μ) |
| (hi : (∫ x, f x ∂μ) ≤ ∫ x, g x ∂μ) {P : α → Prop} (hP : ∀ᵐ x ∂μ, P x) : |
| ∃ x, P x ∧ f x ≤ g x := by |
| classical |
| by_contra h |
| push Not at h |
| have hlt : ∀ᵐ x ∂μ, g x < f x := hP.mono fun x hx => h x hx |
| have hnonneg : 0 ≤ᵐ[μ] fun x => f x - g x := hlt.mono fun x hx => sub_nonneg.mpr hx.le |
| have hzero : (∫ x, f x - g x ∂μ) = 0 := by |
| apply le_antisymm |
| · rw [integral_sub hf hg] |
| linarith |
| · exact integral_nonneg_of_ae hnonneg |
| have hae := (integral_eq_zero_iff_of_nonneg_ae hnonneg (hf.sub hg)).mp hzero |
| have : NeZero μ := ⟨hμ⟩ |
| obtain ⟨x,hx,he⟩ := (hlt.and hae).exists |
| exact (sub_pos.mpr hx).ne' he |
|
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| lemma integral_radial_area (n : ℕ) : |
| (∫ t in Ioo (-1:ℝ) 0, ((n:ℝ)+1)*omega (n+1)*(-t)^n) = omega (n+1) := by |
| let F : ℝ → ℝ := fun t => -omega (n+1)*(-t)^(n+1) |
| let f : ℝ → ℝ := fun t => ((n:ℝ)+1)*omega (n+1)*(-t)^n |
| have hc : Continuous f := by unfold f; fun_prop |
| have hd (t : ℝ) : HasDerivAt F (f t) t := by |
| have hh := (((hasDerivAt_id t).neg.pow (n+1)).const_mul (-omega (n+1))) |
| convert hh using 1 <;> first | rfl | simp only [f, Nat.add_sub_cancel, Nat.cast_add, Nat.cast_one, Pi.neg_apply, id_eq]; ring |
| have hh := intervalIntegral.integral_eq_sub_of_hasDerivAt |
| (fun t _ => hd t) (hc.intervalIntegrable (-1) 0) |
| rw [intervalIntegral.integral_of_le (by norm_num : (-1:ℝ) ≤ 0)] at hh |
| rw [Measure.restrict_congr_set Ioo_ae_eq_Ioc] |
| change (∫ t in Ioc (-1:ℝ) 0, f t) = _ |
| rw [hh] |
| simp [F] |
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| theorem exists_good_inner_ball (n : ℕ) : |
| ∃ t ∈ Ioo (-1:ℝ) 0, |
| IsIntegral (n+1) (innerBallCurrent (n+1) t) ∧ |
| mass (boundarySucc (innerBallCurrent (n+1) t)) ≤ |
| ((n:ℝ)+1)*omega (n+1)*(-t)^n := by |
| obtain ⟨G,hG,hG0,hGI,hgood⟩ := innerBall_ae_integral n |
| let μ := volume.restrict (Ioo (-1:ℝ) 0) |
| let f : ℝ → ℝ := fun t => ((n:ℝ)+1)*omega (n+1)*(-t)^n |
| have hμ : μ ≠ 0 := by |
| intro h |
| have hh := congrArg (fun m : Measure ℝ => m univ) h |
| norm_num [μ] at hh |
| have hf : Integrable f μ := by |
| apply Continuous.integrableOn_Icc (a := (-1:ℝ)) (b := 0) (by unfold f; fun_prop) |>.mono_set |
| exact Ioo_subset_Icc_self |
| have hi : (∫ t, G t ∂μ) ≤ ∫ t, f t ∂μ := by |
| calc (∫ t, G t ∂μ) ≤ ∫ t, G t := |
| setIntegral_le_integral hG (Eventually.of_forall hG0) |
| _ ≤ omega (n+1) := hGI |
| _ = ∫ t, f t ∂μ := (integral_radial_area n).symm |
| obtain ⟨t,⟨ht,htI,htM⟩,hle⟩ := exists_le_of_integral_le_ae hμ hG.restrict hf hi |
| ((ae_restrict_mem measurableSet_Ioo).and hgood) |
| exact ⟨t,ht,htI,htM.trans hle⟩ |
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| end SharpIntegralFillings.Optimality |
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| end |
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| end OAI |
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