File size: 2,121 Bytes
110f4f3 | 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 | import Mathlib
import OAI.Analysis.BiholderTransport.Model
namespace OAI
section
section
noncomputable section
open Set Filter Manifold MeasureTheory Bundle Metric
open scoped Topology ContDiff ENNReal NNReal
namespace WeakMTWTransport
section ForwardChart
variable {n : ℕ} {M : Type*} [MetricSpace M]
[ChartedSpace (Model n) M] [IsManifold 𝓘(ℝ,Model n) ∞ M]
[RiemannianBundle (fun x : M => TangentSpace 𝓘(ℝ,Model n) x)]
[IsContMDiffRiemannianBundle 𝓘(ℝ,Model n) ∞ (Model n)
(fun x : M => TangentSpace 𝓘(ℝ,Model n) x)]
[IsRiemannianManifold 𝓘(ℝ,Model n) M]
attribute [local instance] normedAddCommGroupTangentSpaceVectorSpace normedSpaceTangentSpaceVectorSpace
end ForwardChart
end WeakMTWTransport
end
end
section
noncomputable section
open Set Filter Manifold MeasureTheory Bundle Metric
open scoped Topology ContDiff ENNReal NNReal
namespace WeakMTWTransport
lemma metricVolume_model (n : ℕ) : metricVolume (M := Model n) n=volume := by
let μ : Measure (Model n) := Measure.hausdorffMeasure (n:ℝ)
have hdim : Module.finrank ℝ (Model n)=n := by simp [Model]
have : μ.IsAddHaarMeasure := by
dsimp only [μ]
rw [←hdim]
infer_instance
change _ • μ=volume
exact (measure_eq_sub_vadd volume μ
(measure_ball_pos μ (0:Model n) zero_lt_one).ne' measure_ball_lt_top.ne).symm
lemma metricVolume_lipschitz_image_le {n : ℕ} {X Y : Type*}
[MetricSpace X] [MetricSpace Y] [MeasurableSpace X] [MeasurableSpace Y]
[BorelSpace X] [BorelSpace Y] {f : X → Y} {s : Set X} {C : ℝ≥0}
(hf : LipschitzOnWith C f s) :
metricVolume n (f '' s)≤(C:ℝ≥0∞)^(n:ℝ)*metricVolume n s := by
have H := hf.hausdorffMeasure_image_le (d := (n:ℝ)) (by positivity)
change euclideanVolumeFactor n*(Measure.hausdorffMeasure (n:ℝ) (f '' s))≤
(C:ℝ≥0∞)^(n:ℝ)*(euclideanVolumeFactor n*(Measure.hausdorffMeasure (n:ℝ) s))
calc
_ ≤ euclideanVolumeFactor n*((C:ℝ≥0∞)^(n:ℝ)*(Measure.hausdorffMeasure (n:ℝ) s)) := mul_le_mul_right H _
_ = _ := by ac_rfl
end WeakMTWTransport
end
end
end
end OAI
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