| 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 |
| |
