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namespace OAI
noncomputable section
open Set MeasureTheory Manifold Bundle
open scoped ContDiff Manifold ENNReal NNReal Topology
namespace WeakMTWTransport
abbrev Model (n : ℕ) := EuclideanSpace ℝ (Fin n)
section MetricDefinitions
variable {M : Type*} [MetricSpace M]
/-- Squared-distance cost with factor one half. -/
def cost (x y : M) : ℝ := dist x y ^ 2 / 2
/-- Calibration of Hausdorff measure to Euclidean volume. -/
def euclideanVolumeFactor (n : ℕ) : ℝ≥0∞ :=
volume (Metric.ball (0 : Model n) 1) /
(Measure.hausdorffMeasure (n : ℝ) : Measure (Model n))
(Metric.ball (0 : Model n) 1)
variable [MeasurableSpace M]
/-- Riemannian volume calibrated to Lebesgue measure in Euclidean space. -/
def metricVolume [BorelSpace M] (n : ℕ) : Measure M :=
euclideanVolumeFactor n • Measure.hausdorffMeasure (n : ℝ)
/-- Almost-everywhere measurable probability densities between fixed bounds. -/
def AdmissibleDensity (vol : Measure M) (lam cap : ℝ) (rho : M → ℝ) : Prop :=
AEMeasurable rho vol ∧ Integrable rho vol ∧
(∫ x, rho x ∂vol) = 1 ∧ (∀ᵐ x ∂vol, lam ≤ rho x ∧ rho x ≤ cap)
/-- Multiplication by a density, ignoring exceptional negative values on null sets. -/
def densityMeasure (vol : Measure M) (rho : M → ℝ) : Measure M :=
vol.withDensity (fun x => ENNReal.ofReal (rho x))
/-- Optimality for the Monge minimization problem. -/
def IsOptimalMap (vol : Measure M) (rho0 rho1 : M → ℝ) (T : M → M) : Prop :=
Measurable T ∧ Measure.map T (densityMeasure vol rho0) = densityMeasure vol rho1 ∧
∀ S : M → M, Measurable S →
Measure.map S (densityMeasure vol rho0) = densityMeasure vol rho1 →
(∫ x, cost x (T x) * rho0 x ∂vol) ≤
(∫ x, cost x (S x) * rho0 x ∂vol)
/-- The two global estimates have the very same exponent and constant. -/
def BiHolderEstimate (alpha C : ℝ) (T : M ≃ₜ M) : Prop :=
(∀ x x', dist (T x) (T x') ≤ C * (dist x x') ^ alpha) ∧
(∀ y y', dist (T.symm y) (T.symm y') ≤ C * (dist y y') ^ alpha)
end MetricDefinitions
section RiemannianDefinitions
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)]
/-- Complete affinely parametrized geodesic with prescribed initial tangent.
Local metric minimization is an intrinsic definition of Riemannian geodesics. -/
def IsGeodesicWithInitialData (x : M) (v : TangentSpace 𝓘(ℝ, Model n) x)
(gamma : ℝ → M) : Prop :=
ContMDiff 𝓘(ℝ, ℝ) 𝓘(ℝ, Model n) ∞ gamma ∧
(⟨gamma 0, mfderiv 𝓘(ℝ, ℝ) 𝓘(ℝ, Model n) gamma 0 (1 : ℝ)⟩ :
TangentBundle 𝓘(ℝ, Model n) M) = ⟨x, v⟩ ∧
∀ t : ℝ, ∃ eps : ℝ, 0 < eps ∧
∀ s u : ℝ, |s - t| < eps → |u - t| < eps →
dist (gamma s) (gamma u) = |s - u| * ‖v‖
/-- Evaluation at time one of the geodesic determined by initial data. -/
def riemannianExp (x : M) (v : TangentSpace 𝓘(ℝ, Model n) x) : M := by
classical
exact if h : ∃ gamma, IsGeodesicWithInitialData x v gamma then (Classical.choose h) 1 else x
/-- The open injectivity domain, defined by extendible minimizing vectors. -/
def injectivityDomain (x : M) : Set (TangentSpace 𝓘(ℝ, Model n) x) :=
{p | ∃ a : ℝ, 1 < a ∧ dist x (riemannianExp x (a • p)) = a * ‖p‖}
/-- The Ma–Trudinger–Wang fourth derivative with factor -3/2. -/
def mtw (x : M) (p xi eta : TangentSpace 𝓘(ℝ, Model n) x) : ℝ :=
-(3 / 2 : ℝ) *
iteratedDeriv 2 (fun s : ℝ =>
iteratedDeriv 2 (fun t : ℝ =>
cost (riemannianExp x (t • xi)) (riemannianExp x (p + s • eta))) 0) 0
/-- Weak MTW only at interior velocities and only for orthogonal test vectors. -/
def WeakMTW : Prop :=
∀ (x : M) (p : TangentSpace 𝓘(ℝ, Model n) x), p ∈ injectivityDomain x →
∀ xi eta : TangentSpace 𝓘(ℝ, Model n) x,
inner ℝ xi eta = 0 → 0 ≤ mtw x p xi eta
end RiemannianDefinitions
/-- Uniform bi-Hölder optimal transport for all densities between fixed bounds
on a compact connected smooth Riemannian manifold satisfying weak MTW. -/
theorem uniform_biHolder_transport
{n : ℕ} (hn : 2 ≤ n) {M : Type*}
[MetricSpace M] [CompactSpace M] [ConnectedSpace 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]
[MeasurableSpace M] [BorelSpace M]
(hmtw : WeakMTW (n := n) (M := M))
(lam cap : ℝ) (hlam : 0 < lam) (hcap : lam ≤ cap)
(hclass : ∃ rho : M → ℝ, AdmissibleDensity (metricVolume n) lam cap rho) :
∃ alpha C : ℝ, 0 < alpha ∧ alpha ≤ 1 ∧ 0 ≤ C ∧
∀ rho0 rho1 : M → ℝ,
AdmissibleDensity (metricVolume n) lam cap rho0 →
AdmissibleDensity (metricVolume n) lam cap rho1 →
∃ T : M ≃ₜ M,
IsOptimalMap (metricVolume n) rho0 rho1 T ∧
(∀ S : M → M, IsOptimalMap (metricVolume n) rho0 rho1 S →
S =ᵐ[densityMeasure (metricVolume n) rho0] T) ∧
BiHolderEstimate alpha C T := by
sorry
end WeakMTWTransport
end
end OAI
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