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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 | import Mathlib
import OAI.Geometry.IntegralFillings.Charts.Scalar
import OAI.Geometry.IntegralFillings.Charts.LinearMass
import OAI.Geometry.IntegralFillings.Geometry.Comparison
import OAI.Geometry.IntegralFillings.Geometry.Determinant
import OAI.Geometry.IntegralFillings.Charts.CurrentAxioms
import OAI.Geometry.IntegralFillings.Currents.IntegralPush
namespace OAI
section
open Set Filter MeasureTheory
open scoped Topology ENNReal NNReal RealInnerProductSpace
namespace SharpIntegralFillings
namespace Optimality
lemma innerSpace_CAT0 {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] :
IsCAT0 E := by
let s : E → E → ℝ → E := fun x y t => (1-t) • x + t • y
refine ⟨s, ?_, ?_, ?_⟩
· intro x y
simp [s]
· intro x y a b ha hb
have he : s x y a - s x y b = (a-b) • (y-x) := by dsimp [s]; module
simp only [dist_eq_norm, he, norm_smul, Real.norm_eq_abs]
rw [norm_sub_rev y x]
· intro o x y a b ha hb
have he : s o x a - s o y b = a • (x-o) - b • (y-o) := by dsimp [s]; module
have hi : ‖a • (x-o) - b • (y-o)‖^2 =
a^2 * ‖x-o‖^2 + b^2 * ‖y-o‖^2 -
a*b*(‖x-o‖^2 + ‖y-o‖^2 - ‖(x-o)-(y-o)‖^2) := by
simp only [norm_sub_sq_real, norm_smul, mul_pow, Real.norm_eq_abs, sq_abs,
inner_smul_left, inner_smul_right, starRingEnd_apply, star_trivial]
ring
have hxy : (x-o)-(y-o) = x-y := by abel
rw [hxy] at hi
simp only [dist_eq_norm, he]
rw [hi, norm_sub_rev o x, norm_sub_rev o y]
nlinarith
abbrev UnitBall (d : ℕ) := Metric.closedBall (0 : Euc d) 1
instance (d : ℕ) : Nonempty (UnitBall d) := ⟨⟨0, by simp⟩⟩
lemma unitBall_CAT0 (d : ℕ) : IsCAT0 (UnitBall d) :=
innerSpace_CAT0.closedBall 0 (by norm_num)
noncomputable def ballChart (d : ℕ) : IntegerChart (UnitBall d) d where
domain := Metric.ball 0 1
borel := measurableSet_ball
bounded := Metric.isBounded_ball
param := fun z => ⟨z.val, show dist z.val 0 ≤ 1 from le_of_lt z.property⟩
bilipschitz := ⟨1, 1, by
apply LipschitzWith.of_dist_le_mul
intro x y
simp only [NNReal.coe_one, one_mul]
rfl, by
intro x y
simp only [ENNReal.coe_one, one_mul]
rfl⟩
multiplicity := fun _ => 1
integrable := by
simp only [Int.cast_one]
exact integrableOn_const (μ := volume) (C := (1:ℝ)) Metric.isBounded_ball.measure_lt_top.ne
lemma chart_integerRectifiable {X : Type*} [MetricSpace X] [MeasurableSpace X]
[BorelSpace X] {d : ℕ} (C : IntegerChart X d) : IntegerRectifiable C.action := by
classical
let D : ℕ → IntegerChart X d := fun i => if i = 0 then C else emptyChart d
have hD (i : ℕ) : IsMetricCurrent (D i).action := (D i).action_isMetricCurrent
refine ⟨D, ?_, hD, ?_, ?_⟩
· intro i j hij
by_cases hi : i = 0
· have hj : j ≠ 0 := by omega
simp [D, hj]
· simp [D, hi]
· apply summable_of_ne_finset_zero (s := {0})
intro i hi
have hi' : i ≠ 0 := by simpa using hi
simp [D, hi']
· intro b π
rw [tsum_eq_single 0]
· simp [D]
· intro i hi
simp [D, hi]
lemma det_le_one_of_clm_norm {d : ℕ} (p : Fin d → Euc d →L[ℝ] ℝ)
(hp : ∀ i, ‖p i‖ ≤ 1) :
|Matrix.det (fun i j => p i (EuclideanSpace.single j 1))| ≤ 1 := by
apply abs_det_le_one_of_dot_bound
intro i v
have hv : ∑ j, v j • EuclideanSpace.single j (1:ℝ) =
(WithLp.toLp 2 v : Euc d) := by
convert (EuclideanSpace.basisFun (Fin d) ℝ).sum_repr (WithLp.toLp 2 v) using 1
simp
have he : (fun j => p i (EuclideanSpace.single j 1)) ⬝ᵥ v =
p i (WithLp.toLp 2 v) := by
rw [← hv, map_sum]
simp only [map_smul, smul_eq_mul, dotProduct]
apply Finset.sum_congr rfl
intro j hj
exact mul_comm _ _
rw [he]
exact (p i).le_opNorm _ |>.trans (by simpa using mul_le_mul_of_nonneg_right (hp i) (norm_nonneg (WithLp.toLp 2 v : Euc d)))
lemma chart_mass_le_unitLip {X : Type*} [MetricSpace X] [MeasurableSpace X]
[BorelSpace X] [Nonempty X] {d : ℕ} (C : IntegerChart X d)
(hC : LipschitzWith 1 C.param) :
mass C.action ≤ ∫ z in C.domain, |(C.multiplicity z : ℝ)| := by
let μ := C.majorantMeasure (fun _ => 1)
have hρ : Integrable (fun z => |(C.multiplicity z : ℝ)| * 1)
(volume.restrict C.domain) := by simpa using C.integrable.abs
have hfin : IsFiniteMeasure μ := densityPush_finite _ _ hρ
have hctrl : Controls C.action μ := by
apply C.majorant_controls hρ (Eventually.of_forall fun _ => zero_le_one)
intro π hπ
have hnorm : ∀ᵐ z ∂volume.restrict C.domain, ∀ i,
‖fderivWithin ℝ (C.scalar (π i)) C.domain z‖ ≤ 1 := by
apply ae_all_iff.mpr
intro i
have hs : LipschitzOnWith 1 (C.scalar (π i)) C.domain := by
simpa only [one_mul] using C.scalar_lipschitzOn hC (hπ i)
simpa using ae_norm_fderivWithin_le volume C.borel hs
filter_upwards [hnorm] with z hz
exact det_le_one_of_clm_norm _ hz
apply (mass_le_measure hfin hctrl).trans_eq
have hn : 0 ≤ᵐ[volume.restrict C.domain] fun z => |(C.multiplicity z : ℝ)| * 1 :=
Eventually.of_forall fun z => by positivity
have h := integral_densityPush (volume.restrict C.domain) C.measurable_paramExtended
hρ hn (continuous_const : Continuous (fun _ : X => (1:ℝ)))
simpa only [integral_const, smul_eq_mul, mul_one, IntegerChart.majorantMeasure, μ] using h
lemma ballChart_mass_le (d : ℕ) : mass (ballChart d).action ≤ omega d := by
have hL : LipschitzWith 1 (ballChart d).param := by
apply LipschitzWith.of_dist_le_mul
intro x y
simp only [NNReal.coe_one, one_mul]
rfl
have h := chart_mass_le_unitLip (ballChart d) hL
simpa [ballChart, omega, Measure.real] using h
end Optimality
end SharpIntegralFillings
end
end OAI
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