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59d1c80 | 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 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 | import Mathlib
import OAI.Geometry.IntegralFillings.Optimality.BallCoarea
namespace OAI
section
open Set Filter MeasureTheory
open scoped Topology ENNReal NNReal
namespace SharpIntegralFillings.Optimality
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
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]
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⟩
end SharpIntegralFillings.Optimality
end
end OAI
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