| import OAI.Geometry.RCD.Transport.OptimalDualSaturation |
| import OAI.Geometry.RCD.Transport.CompactCouplingAmbientCost |
|
|
| namespace OAI |
|
|
| /-! # The same dual pair calibrates every ambient optimal coupling -/ |
|
|
| open Set Filter MeasureTheory |
| open scoped ENNReal |
| noncomputable section |
| namespace RegularPointCoordinates |
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] |
| [SecondCountableTopology X] {S T : Set X} [CompactSpace S] [CompactSpace T] |
|
|
| theorem ambient_optimal_dual_saturation (hS : MeasurableSet S) (hT : MeasurableSet T) |
| (μ : Measure S) (ν : Measure T) [IsProbabilityMeasure μ] [IsProbabilityMeasure ν] |
| (Q : Measure (S × T)) [IsProbabilityMeasure Q] |
| (hQl : Q.map Prod.fst=μ) (hQr : Q.map Prod.snd=ν) |
| (f : C(S,ℝ)) (g : C(T,ℝ)) (c : ℝ) |
| (hdual : ∀ a b, b ∈ ν.support → f a+g b+c ≤ dist (a : X) (b : X)^2) |
| (heq : ∀ᵐ z ∂Q, f z.1+g z.2+c=dist (z.1 : X) (z.2 : X)^2) |
| (P : Measure (X × X)) (hP : IsOptimalCoupling (μ.map Subtype.val) (ν.map Subtype.val) P) : |
| ∃ L : Measure (S × T), L.map (Prod.map Subtype.val Subtype.val)=P ∧ |
| L.map Prod.fst=μ ∧ L.map Prod.snd=ν ∧ |
| ∀ᵐ z ∂L, f z.1+g z.2+c=dist (z.1 : X) (z.2 : X)^2 := by |
| let cost : C(S × T,ℝ) := ⟨fun z => dist (z.1 : X) (z.2 : X)^2,by fun_prop⟩ |
| have hmin := compact_dual_cost_minimizer μ ν Q hQl hQr f g cost c hdual heq |
| have hQ := compact_coupling_ambient_optimal hS hT μ ν Q hQl hQr hmin |
| obtain ⟨L,hL,hLl,hLr⟩ := coupling_lift_subtypes hS hT μ ν P hP.1 hP.2.1 |
| let : IsProbabilityMeasure (L.map Prod.fst) := by rw [hLl]; infer_instance |
| let : IsProbabilityMeasure L := Measure.isProbabilityMeasure_of_map measurable_fst.aemeasurable |
| have hle := hP.2.2 _ hQ.1 hQ.2.1 |
| rw [← hL,compact_coupling_transportCost L,compact_coupling_transportCost Q] at hle |
| have hcost : (∫ z, cost z ∂L) ≤ ∫ z, cost z ∂Q := |
| (ENNReal.ofReal_le_ofReal_iff (integral_nonneg fun _ => sq_nonneg _)).mp hle |
| exact ⟨L,hL,hLl,hLr,compact_dual_saturation_of_cost_le μ ν Q L hQl hQr hLl hLr |
| f g cost c hdual heq hcost⟩ |
|
|
| end RegularPointCoordinates |
|
|
| end |
|
|
| end OAI |
|
|
