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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 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 | import Mathlib
import OAI.Geometry.IntegralFillings.Optimality.BallCoarea
namespace OAI
section
open Set Filter MeasureTheory
open scoped Topology ENNReal NNReal
namespace SharpIntegralFillings.Optimality
open BorelCoefficients BorelRestriction MassMeasure CurrentOperations
noncomputable def ambientCoord {d : ℕ} (i : Fin d) : Euc d → ℝ := EuclideanSpace.proj i
noncomputable def unitCoord {d : ℕ} (i : Fin d) (x : UnitBall d) : ℝ := ambientCoord i x.val
lemma ambientCoord_lipschitz {d : ℕ} (i : Fin d) : LipschitzWith 1 (ambientCoord i) := by
apply LipschitzWith.of_dist_le_mul
intro x y
simp only [NNReal.coe_one, one_mul, Real.dist_eq]
rw [dist_eq_norm]
change ‖(x-y) i‖ ≤ ‖x-y‖
exact PiLp.norm_apply_le (x-y) i
lemma unitCoord_lipschitz {d : ℕ} (i : Fin d) : LipschitzWith 1 (unitCoord i) := by
apply LipschitzWith.of_dist_le_mul
intro x y
exact (ambientCoord_lipschitz i).dist_le_mul x.val y.val
lemma unitCoord_bound {d : ℕ} (i : Fin d) (x : UnitBall d) : |unitCoord i x| ≤ 1 := by
have hx : ‖x.val‖ ≤ 1 := by simpa only [Metric.mem_closedBall, dist_zero_right] using x.property
exact (PiLp.norm_apply_le x.val i).trans hx
lemma unitCoord_boundedLip {d : ℕ} (i : Fin d) : BoundedLip (unitCoord i) :=
⟨⟨1,unitCoord_lipschitz i⟩,1,unitCoord_bound i⟩
lemma ballChart_jacobian_coords (d : ℕ) :
(ballChart d).jacobian (fun i => unitCoord i) =ᵐ[volume.restrict (ballChart d).domain]
fun _ => 1 := by
have he (i : Fin d) : EqOn ((ballChart d).scalar (unitCoord i)) (ambientCoord i)
(ballChart d).domain := by
intro z hz
rw [(ballChart d).scalar_eq hz]
rfl
filter_upwards [(ballChart d).ae_jacobian_eq_extensions ambientCoord_lipschitz he] with z hz
rw [hz]
have hM : (Matrix.of fun (i j : Fin d) =>
fderiv ℝ (ambientCoord i) z (EuclideanSpace.single j 1)) = 1 := by
ext i j
change (fderiv ℝ (EuclideanSpace.proj i) z) (EuclideanSpace.single j 1) = _
rw [ContinuousLinearMap.fderiv]
simp [Matrix.one_apply]
rw [hM,Matrix.det_one]
lemma innerBallCurrent_coords (d : ℕ) {t : ℝ} (ht : -1 < t) :
innerBallCurrent d t (fun _ => 1) (fun i => unitCoord i) =
(volume (Metric.ball (0 : Euc d) (-t))).toReal := by
let C := ballChart d
let E := {x : UnitBall d | t < radiusFunction d x}
have hE : MeasurableSet E := measurableSet_lt measurable_const
(radiusFunction_lipschitz d).continuous.measurable
have had : Admissible (fun _ : UnitBall d => (1:ℝ)) (fun i => unitCoord i) :=
⟨BoundedLip.const 1,fun i => ⟨1,unitCoord_lipschitz i⟩⟩
rw [innerBallCurrent, restrictCurrent_apply C.action_isMetricCurrent E had]
rw [C.borelAction_eq_integral C.action_isMetricCurrent
(currentMassMeasure_controls C.action_isMetricCurrent)
(measurable_const.indicator hE) (show ∃ M : ℝ, ∀ x, |E.indicator (fun _ => (1:ℝ)) x| ≤ M from
⟨1,fun x => by by_cases hx : x ∈ E <;> simp [hx]⟩)
(fun i => unitCoord i) had.2]
calc (∫ z in C.domain, (C.multiplicity z : ℝ) * C.jacobian (fun i => unitCoord i) z *
E.indicator (fun _ => (1:ℝ)) (C.paramExtended z)) =
∫ z in Metric.ball (0 : Euc d) 1,
(Metric.ball (0 : Euc d) (-t)).indicator (fun _ => (1:ℝ)) z := by
apply integral_congr_ae
filter_upwards [ballChart_jacobian_coords d, ae_restrict_mem C.borel] with z hj hz
have hp : radiusFunction d (C.paramExtended z) = -‖z‖ := by
have hz' : z ∈ Metric.ball (0 : Euc d) 1 := hz
simp [IntegerChart.paramExtended, hz', C, ballChart, radiusFunction]
have he : C.paramExtended z ∈ E ↔ z ∈ Metric.ball (0 : Euc d) (-t) := by
change t < radiusFunction d (C.paramExtended z) ↔ dist z 0 < -t
rw [hp,dist_zero_right]
constructor <;> intro h <;> linarith
change (1:ℤ) * C.jacobian (fun i => unitCoord i) z * _ = _
rw [Int.cast_one, hj, one_mul, one_mul]
by_cases hh : z ∈ Metric.ball (0 : Euc d) (-t)
· simp [hh,he.mpr hh]
· simp [hh,not_congr he |>.mpr hh]
_ = (volume (Metric.ball (0 : Euc d) (-t))).toReal := by
rw [integral_indicator measurableSet_ball]
simp only [integral_const, smul_eq_mul, mul_one, Measure.real,
Measure.restrict_apply_univ, Measure.restrict_apply measurableSet_ball]
rw [inter_eq_left.mpr (Metric.ball_subset_ball (show -t ≤ 1 by linarith))]
lemma omega_pos (d : ℕ) : 0 < omega d := by
exact ENNReal.toReal_pos (Metric.measure_ball_pos volume (0 : Euc d) (by norm_num)).ne'
Metric.isBounded_ball.measure_lt_top.ne
lemma ball_volume_scaling (d : ℕ) {r : ℝ} (hr : 0 < r) :
(volume (Metric.ball (0 : Euc d) r)).toReal = omega d * r^d := by
rw [Measure.addHaar_ball_of_pos volume 0 hr, ENNReal.toReal_mul,
ENNReal.toReal_ofReal (pow_nonneg hr.le _)]
simp only [finrank_euclideanSpace, Fintype.card_fin, omega]
ring
lemma abs_apply_le_mass {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X]
{k : ℕ} {T : Functional X k} (hT : IsMetricCurrent T) {b : X → ℝ}
{π : Fin k → X → ℝ} (hb : BoundedLip b) (hb1 : ∀ x, |b x| ≤ 1)
(hπ : ∀ i, LipschitzWith 1 (π i)) : |T b π| ≤ mass T := by
apply le_csInf
· obtain ⟨μ,hμ,hctrl⟩ := hT.finiteMass
exact ⟨μ.real univ, μ,hμ,hctrl,rfl⟩
· rintro r ⟨μ,hμ,hctrl,rfl⟩
let := hμ
apply (hctrl b π hb hπ).trans
calc (∫ x, |b x| ∂μ) ≤ ∫ _ : X, (1:ℝ) ∂μ :=
integral_mono ((integrable_const (1:ℝ)).mono' hb.continuous.aestronglyMeasurable
(Eventually.of_forall fun x => by simpa only [Real.norm_eq_abs] using hb1 x)).abs
(integrable_const 1) hb1
_ = μ.real univ := by simp
end SharpIntegralFillings.Optimality
end
end OAI
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