import OAI.Geometry.RCD.Transport.AsymmetricMidpointReplacement namespace OAI /-! # An actual compressed midpoint with unequal endpoint bounds Minimizing excess mass and replacing its saturated region turns the CD support estimate into a density bound. Separate endpoint bounds allow one marginal to concentrate while retaining the contribution of the other. -/ open Set Filter Metric MeasureTheory TopologicalSpace open scoped ENNReal namespace RegularPointCoordinates variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] attribute [local instance] pathBorel pathBorelSpace theorem FullCD.improve_localized_saturation_asymmetric (μ : Measure X) [SigmaFinite μ] (W : Set X) (hW : MeasurableSet W) [IsFiniteMeasure (μ.restrict W)] (K : ℝ) (n : ℕ) (hcd : FullCD μ K n) (ν₀ ν₁ : Measure X) [IsProbabilityMeasure ν₀] [IsProbabilityMeasure ν₁] {C₀ C₁ C : ℝ≥0∞} (hC₀ : C₀ ≠ ∞) (hC₁ : C₁ ≠ ∞) (hC : C ≠ ∞) (hCpos : C ≠ 0) (h0 : ν₀ ≤ C₀ • μ) (h1 : ν₁ ≤ C₁ • μ) (hm0 : FiniteSecondMoment ν₀) (hm1 : FiniteSecondMoment ν₁) (s₀ s₁ : Set X) (hs0 : ∀ᵐ x ∂ν₀, x ∈ s₀) (hs1 : ∀ᵐ x ∂ν₁, x ∈ s₁) (hn : 2 ≤ n) (D : ℝ) (hD : ∀ x ∈ s₀, ∀ y ∈ s₁, dist x y ≤ D) (hsupport : ∀ (q : Measure C(Time, X)), IsProbabilityMeasure q → (∀ᵐ γ ∂q, IsGeodesic γ) → IsOptimalCoupling ν₀ ν₁ (q.map (fun γ => (γ 0, γ 1))) → ∀ᵐ x ∂q.map (fun γ => γ midpointTime), x ∈ W) (hfill : C⁻¹ ≤ midpointSupportFactor K n D C₀ C₁) (π : Measure C(Time, X)) [IsProbabilityMeasure π] (hg : ∀ᵐ γ ∂π, IsGeodesic γ) (hp : IsOptimalCoupling ν₀ ν₁ (π.map (fun γ => (γ 0, γ 1)))) (hpos : transportExcessMass (μ.restrict W) (π.map (fun γ => γ midpointTime)) C ≠ 0) (hmin : ∀ (π' : Measure C(Time, X)), IsProbabilityMeasure π' → (∀ᵐ γ ∂π', IsGeodesic γ) → IsOptimalCoupling ν₀ ν₁ (π'.map (fun γ => (γ 0, γ 1))) → transportExcessMass (μ.restrict W) (π.map (fun γ => γ midpointTime)) C ≤ transportExcessMass (μ.restrict W) (π'.map (fun γ => γ midpointTime)) C) : ∃ π' : Measure C(Time, X), IsProbabilityMeasure π' ∧ (∀ᵐ γ ∂π', IsGeodesic γ) ∧ IsOptimalCoupling ν₀ ν₁ (π'.map (fun γ => (γ 0, γ 1))) ∧ transportExcessMass (μ.restrict W) (π'.map (fun γ => γ midpointTime)) C = transportExcessMass (μ.restrict W) (π.map (fun γ => γ midpointTime)) C ∧ (μ.restrict W) {x | C < (π.map (fun γ => γ midpointTime)).rnDeriv (μ.restrict W) x} + transportExcessMass (μ.restrict W) (π.map (fun γ => γ midpointTime)) C / 2 * C⁻¹ < (μ.restrict W) {x | C < (π'.map (fun γ => γ midpointTime)).rnDeriv (μ.restrict W) x} := by let ν := π.map (fun γ => γ midpointTime) have hvprob : IsProbabilityMeasure ν := inferInstance let e := transportExcessMass (μ.restrict W) ν C let v := (μ.restrict W) {x | C < ν.rnDeriv (μ.restrict W) x} have hetop : e ≠ ∞ := ne_top_of_le_ne_top (measure_ne_top ν univ) (transportExcessMass_le (μ.restrict W) ν C) obtain ⟨s, hs, hμs, hνs⟩ := (Measure.mutuallySingular_singularPart ν (μ.restrict W)).symm have hthreshold : e / 2 + C * v < ν (s ∪ {x | C < ν.rnDeriv (μ.restrict W) x}) := by rw [positive_region_mass (μ.restrict W) ν hC hs hμs hνs] exact ENNReal.add_lt_add_right (ENNReal.mul_ne_top hC (measure_ne_top (μ.restrict W) _)) (ENNReal.half_lt_self hpos hetop) obtain ⟨δ, hδ0, hδtop, hδmass⟩ := exists_buffer_positive_region (μ.restrict W) ν hC s hthreshold let B := s ∪ {x | C + δ < ν.rnDeriv (μ.restrict W) x} have hB : MeasurableSet B := hs.union (measurableSet_lt measurable_const (ν.measurable_rnDeriv (μ.restrict W))) have hB0 : ν B ≠ 0 := (lt_of_le_of_lt zero_le hδmass).ne' let a := δ / (C + δ) have ha : a ≤ 1 := ENNReal.div_le_of_le_mul (by simp) have ha0 : 0 < a := ENNReal.div_pos hδ0.ne' (ENNReal.add_ne_top.mpr ⟨hC, hδtop⟩) have hrem : a • ν.restrict B ≤ ν - C • μ.restrict W := buffer_fraction_le_excess (μ.restrict W) ν hC hδ0 hδtop hs hμs let : IsFiniteMeasure (a • ν.restrict B) := isFiniteMeasure_of_le ν (hrem.trans Measure.sub_le) obtain ⟨π', η, hπ', hη, hg', hp', heq, hmass, hvol⟩ := hcd.replace_midpoint_region_asymmetric μ K n ν₀ ν₁ hC₀ hC₁ h0 h1 hm0 hm1 s₀ s₁ hs0 hs1 hn D hD π hg hp B hB hB0 ha0 ha let := hπ' let := hη have hηle : η ≤ π'.map (fun γ => γ midpointTime) := by rw [heq] exact Measure.le_add_left le_rfl have hηsupp : ∀ᵐ x ∂η, x ∈ W := (hsupport π' hπ' hg' hp').filter_mono hηle.absolutelyContinuous.ae_le have hvolLoc := hvol.trans_eq (ac_support_volume_restrict μ η hW hηsupp) have hmin' := hmin π' hπ' hg' hp' have hmass' : η univ = (a • ν.restrict B) univ := by simpa only [Measure.smul_apply, smul_eq_mul, Measure.restrict_apply_univ] using hmass have hequal : transportExcessMass (μ.restrict W) (π'.map (fun γ => γ midpointTime)) C = e := by apply le_antisymm _ hmin' rw [heq] exact (transportExcessMass_add_le (μ.restrict W) (ν - a • ν.restrict B) η hC).trans_eq (by rw [hmass', transportExcessMass_remove (μ.restrict W) ν (a • ν.restrict B) hC hrem]) have hsaturated : ∀ᵐ x ∂μ.restrict W, η.rnDeriv (μ.restrict W) x ≠ 0 → C < (π'.map (fun γ => γ midpointTime)).rnDeriv (μ.restrict W) x := by rw [heq] apply ac_support_saturated_of_minimizing_replacement (μ.restrict W) ν (a • ν.restrict B) η hC hrem hmass' simpa only [heq] using hmin' refine ⟨π', hπ', hg', hp', hequal, ?_⟩ have hgain : v + e / 2 * C⁻¹ < ν B * C⁻¹ := by have hh := ENNReal.mul_lt_mul_left (ENNReal.inv_ne_zero.mpr hC) (ENNReal.inv_ne_top.mpr hCpos) hδmass convert hh using 1 rw [add_mul, mul_assoc, mul_comm v, ← mul_assoc, ENNReal.mul_inv_cancel hCpos hC] simp only [one_mul, add_comm] exact hgain.trans_le (calc ν B * C⁻¹ ≤ ν B * (midpointSupportFactor K n D C₀ C₁) := mul_le_mul_right hfill _ _ = midpointSupportFactor K n D C₀ C₁ * ν B := by ac_rfl _ ≤ (μ.restrict W) {x | η.rnDeriv (μ.restrict W) x ≠ 0} := hvolLoc _ ≤ (μ.restrict W) {x | C < (π'.map (fun γ => γ midpointTime)).rnDeriv (μ.restrict W) x} := measure_mono_ae hsaturated) theorem FullCD.localized_excess_minimizer_zero_asymmetric (μ : Measure X) [SigmaFinite μ] (W : Set X) (hW : MeasurableSet W) [IsFiniteMeasure (μ.restrict W)] (K : ℝ) (n : ℕ) (hcd : FullCD μ K n) (ν₀ ν₁ : Measure X) [IsProbabilityMeasure ν₀] [IsProbabilityMeasure ν₁] {C₀ C₁ C : ℝ≥0∞} (hC₀ : C₀ ≠ ∞) (hC₁ : C₁ ≠ ∞) (hC : C ≠ ∞) (hCpos : C ≠ 0) (h0 : ν₀ ≤ C₀ • μ) (h1 : ν₁ ≤ C₁ • μ) (hm0 : FiniteSecondMoment ν₀) (hm1 : FiniteSecondMoment ν₁) (s₀ s₁ : Set X) (hs0 : ∀ᵐ x ∂ν₀, x ∈ s₀) (hs1 : ∀ᵐ x ∂ν₁, x ∈ s₁) (hn : 2 ≤ n) (D : ℝ) (hD : ∀ x ∈ s₀, ∀ y ∈ s₁, dist x y ≤ D) (hsupport : ∀ (q : Measure C(Time, X)), IsProbabilityMeasure q → (∀ᵐ γ ∂q, IsGeodesic γ) → IsOptimalCoupling ν₀ ν₁ (q.map (fun γ => (γ 0, γ 1))) → ∀ᵐ x ∂q.map (fun γ => γ midpointTime), x ∈ W) (hfill : C⁻¹ ≤ midpointSupportFactor K n D C₀ C₁) (π : Measure C(Time, X)) [IsProbabilityMeasure π] (hg : ∀ᵐ γ ∂π, IsGeodesic γ) (hp : IsOptimalCoupling ν₀ ν₁ (π.map (fun γ => (γ 0, γ 1)))) (hmin : ∀ (π' : Measure C(Time, X)), IsProbabilityMeasure π' → (∀ᵐ γ ∂π', IsGeodesic γ) → IsOptimalCoupling ν₀ ν₁ (π'.map (fun γ => (γ 0, γ 1))) → transportExcessMass (μ.restrict W) (π.map (fun γ => γ midpointTime)) C ≤ transportExcessMass (μ.restrict W) (π'.map (fun γ => γ midpointTime)) C) : transportExcessMass (μ.restrict W) (π.map (fun γ => γ midpointTime)) C = 0 := by by_contra hpos let e := transportExcessMass (μ.restrict W) (π.map (fun γ => γ midpointTime)) C let A := {ρ : Measure C(Time, X) // IsProbabilityMeasure ρ ∧ (∀ᵐ γ ∂ρ, IsGeodesic γ) ∧ IsOptimalCoupling ν₀ ν₁ (ρ.map (fun γ => (γ 0, γ 1))) ∧ transportExcessMass (μ.restrict W) (ρ.map (fun γ => γ midpointTime)) C = e} have : Nonempty A := ⟨⟨π, inferInstance, hg, hp, rfl⟩⟩ let v (ρ : A) := (μ.restrict W) {x | C < (ρ.val.map (fun γ => γ midpointTime)).rnDeriv (μ.restrict W) x} let S := ⨆ ρ : A, v ρ have hS : S ≠ ∞ := ne_top_of_le_ne_top (measure_ne_top (μ.restrict W) univ) (iSup_le fun _ => measure_mono (subset_univ _)) have hstep : S + e / 2 * C⁻¹ ≤ S := by rw [ENNReal.iSup_add] apply iSup_le intro ρ let : IsProbabilityMeasure ρ.val := ρ.property.1 have hposρ : transportExcessMass (μ.restrict W) (ρ.val.map (fun γ => γ midpointTime)) C ≠ 0 := by rw [ρ.property.2.2.2] exact hpos have hminρ : ∀ (π' : Measure C(Time, X)), IsProbabilityMeasure π' → (∀ᵐ γ ∂π', IsGeodesic γ) → IsOptimalCoupling ν₀ ν₁ (π'.map (fun γ => (γ 0, γ 1))) → transportExcessMass (μ.restrict W) (ρ.val.map (fun γ => γ midpointTime)) C ≤ transportExcessMass (μ.restrict W) (π'.map (fun γ => γ midpointTime)) C := by intro π' hπ' hg' hp' rw [ρ.property.2.2.2] exact hmin π' hπ' hg' hp' obtain ⟨π', hπ', hg', hp', hequal, hgain⟩ := hcd.improve_localized_saturation_asymmetric μ W hW K n ν₀ ν₁ hC₀ hC₁ hC hCpos h0 h1 hm0 hm1 s₀ s₁ hs0 hs1 hn D hD hsupport hfill ρ.val ρ.property.2.1 ρ.property.2.2.1 hposρ hminρ let ρ' : A := ⟨π', hπ', hg', hp', hequal.trans ρ.property.2.2.2⟩ rw [ρ.property.2.2.2] at hgain exact hgain.le.trans (le_iSup v ρ') have hδ : 0 < e / 2 * C⁻¹ := ENNReal.mul_pos (ENNReal.div_pos hpos (by simp)).ne' (ENNReal.inv_ne_zero.mpr hC) have hstrict := ENNReal.add_lt_add_left hS hδ rw [add_zero] at hstrict exact (not_lt_of_ge hstep) hstrict section Attainment variable [ProperSpace X] [SeparableSpace X] theorem FullCD.exists_compressed_midpoint_asymmetric (μ : Measure X) [SigmaFinite μ] (K : ℝ) (n : ℕ) (hcd : FullCD μ K n) (ν₀ ν₁ : Measure X) [IsProbabilityMeasure ν₀] [IsProbabilityMeasure ν₁] {C₀ C₁ C : ℝ≥0∞} (hC₀ : C₀ ≠ ∞) (hC₁ : C₁ ≠ ∞) (hC : C ≠ ∞) (hCpos : C ≠ 0) (h0 : ν₀ ≤ C₀ • μ) (h1 : ν₁ ≤ C₁ • μ) (hm0 : FiniteSecondMoment ν₀) (hm1 : FiniteSecondMoment ν₁) (p : X) {R : ℝ} (hR : 0 ≤ R) [IsFiniteMeasure (μ.restrict (closedBall p (3 * R)))] [Measure.OuterRegular (μ.restrict (closedBall p (3 * R)))] (hs0 : ∀ᵐ x ∂ν₀, dist x p ≤ R) (hs1 : ∀ᵐ x ∂ν₁, dist x p ≤ R) (hn : 2 ≤ n) (hfill : C⁻¹ ≤ midpointSupportFactor K n (2 * R) C₀ C₁) : ∃ π : Measure C(Time, X), IsProbabilityMeasure π ∧ (∀ᵐ γ ∂π, IsGeodesic γ) ∧ IsOptimalCoupling ν₀ ν₁ (π.map (fun γ => (γ 0, γ 1))) ∧ π.map (fun γ => γ midpointTime) ≤ C • μ := by let W := closedBall p (3 * R) have hW : MeasurableSet W := isClosed_closedBall.measurableSet have hsupport : ∀ (q : Measure C(Time, X)), IsProbabilityMeasure q → (∀ᵐ γ ∂q, IsGeodesic γ) → IsOptimalCoupling ν₀ ν₁ (q.map (fun γ => (γ 0, γ 1))) → ∀ᵐ x ∂q.map (fun γ => γ midpointTime), x ∈ W := by intro q hq hgq hpq apply (ae_map_iff (measurable_path_eval midpointTime).aemeasurable hW).mpr exact (optimal_geodesic_plan_bounded ν₀ ν₁ q hgq hpq p hR hs0 hs1).mono (fun γ hγ => hγ.2 midpointTime) let v0 : ProbabilityMeasure X := ⟨ν₀, inferInstance⟩ let v1 : ProbabilityMeasure X := ⟨ν₁, inferInstance⟩ obtain ⟨q, hq, hgq, hpq, _⟩ := hcd ν₀ ν₁ inferInstance inferInstance (Measure.absolutelyContinuous_of_le_smul h0) (Measure.absolutelyContinuous_of_le_smul h1) hm0 hm1 have hne : (OptimalGeodesicPlans v0 v1).Nonempty := ⟨⟨q, hq⟩, hgq, hpq⟩ obtain ⟨π, hπ, hmπ⟩ := exists_optimal_geodesic_plan_minimizing_excess (μ.restrict W) v0 v1 p hR hs0 hs1 hne midpointTime hC have hmin : ∀ (q : Measure C(Time, X)), IsProbabilityMeasure q → (∀ᵐ γ ∂q, IsGeodesic γ) → IsOptimalCoupling ν₀ ν₁ (q.map (fun γ => (γ 0, γ 1))) → transportExcessMass (μ.restrict W) ((π : Measure C(Time, X)).map (fun γ => γ midpointTime)) C ≤ transportExcessMass (μ.restrict W) (q.map (fun γ => γ midpointTime)) C := by intro q hq hgq hpq exact hmπ ⟨q, hq⟩ ⟨hgq, hpq⟩ have hD : ∀ x ∈ closedBall p R, ∀ y ∈ closedBall p R, dist x y ≤ 2 * R := by intro x hx y hy have hd := dist_triangle x p y rw [dist_comm p y] at hd change dist x p ≤ R at hx change dist y p ≤ R at hy linarith have hz := hcd.localized_excess_minimizer_zero_asymmetric μ W hW K n ν₀ ν₁ hC₀ hC₁ hC hCpos h0 h1 hm0 hm1 (closedBall p R) (closedBall p R) hs0 hs1 hn (2 * R) hD hsupport hfill (π : Measure C(Time, X)) hπ.1 hπ.2 hmin refine ⟨π, inferInstance, hπ.1, hπ.2, ?_⟩ have hs := hsupport (π : Measure C(Time, X)) inferInstance hπ.1 hπ.2 have ht := localized_excess_zero_iff μ ((π : Measure C(Time, X)).map (fun γ => γ midpointTime)) (B := W) (C := C) hs hC exact ht.mp hz end Attainment end RegularPointCoordinates end OAI