| import OAI.Geometry.RCD.Transport.RadialMidpoint |
| import OAI.Geometry.RCD.Transport.OptimalRadialMidpoint |
|
|
| namespace OAI |
|
|
| /-! # A bounded-compression radial plan from an actual bounded density |
|
|
| First construct the radial midpoint by endpoint concentration. Apply the |
| proved bounded-compression transport theorem to that midpoint. Optimal |
| cost rigidity forces the replacement to have exactly the radial speed. |
| -/ |
|
|
| open Set Filter Metric MeasureTheory TopologicalSpace |
| open scoped ENNReal |
|
|
| namespace RegularPointCoordinates |
|
|
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] [ProperSpace X] |
| attribute [local instance] pathBorel pathBorelSpace |
|
|
| theorem NoncollapsedRCD.exists_compressed_radial_plan {K : ℝ} {n : ℕ} |
| (hX : NoncollapsedRCD K n X) (hn : 2 ≤ n) (ν₀ : ProbabilityMeasure X) |
| {C₀ : ℝ≥0∞} (hC₀ : C₀ ≠ ∞) (h0 : (ν₀ : Measure X) ≤ C₀ • hausdorff n X) |
| (p : X) {R : ℝ} (hR : 0 < R) |
| (hs0 : ∀ᵐ x ∂(ν₀ : Measure X), dist x p ≤ R) : |
| ∃ π : Measure C(Time, X), IsProbabilityMeasure π ∧ |
| ∃ C : ℝ≥0∞, C ≠ ∞ ∧ |
| π.map (fun γ => γ 0) = (ν₀ : Measure X) ∧ |
| (∀ᵐ γ ∂π, IsGeodesic γ) ∧ |
| MemLp (fun γ : C(Time, X) => dist (γ 0) (γ 1)) 2 π ∧ |
| (∀ t : Time, π.map (fun γ => γ t) ≤ C • hausdorff n X) ∧ |
| ∀ᵐ γ ∂π, dist (γ 0) (γ 1) = dist (γ 1) p ∧ |
| dist (γ 0) p = dist (γ 0) (γ 1) + dist (γ 1) p := by |
| let := hX.separable |
| let := locallyFinite_of_finiteOnBounded (hausdorff n X) hX.finiteOnBounded |
| let μ := hausdorff n X |
| obtain ⟨q, hq, hq0, hq1, hqm⟩ := hX.exists_radial_midpoint_plan hn ν₀ hC₀ h0 p hR hs0 |
| let η : Measure X := (q.map (fun γ => γ midpointTime) : Measure X) |
| let C₁ := C₀ / (midpointFloor K n (2 * R)) ^ n |
| have hC₁ : C₁ ≠ ∞ := |
| ENNReal.div_ne_top hC₀ (pow_ne_zero _ (midpointFloor_pos K n (2 * R)).ne') |
| let A := max C₀ C₁ |
| have hA : A ≠ ∞ := (max_lt_iff.mpr ⟨hC₀.lt_top, hC₁.lt_top⟩).ne |
| have hb0 : (ν₀ : Measure X) ≤ A • μ := by |
| intro S |
| exact (h0 S).trans (mul_le_mul_left (le_max_left C₀ C₁) (μ S)) |
| have hb1 : η ≤ A • μ := by |
| intro S |
| exact (hqm S).trans (mul_le_mul_left (le_max_right C₀ C₁) (μ S)) |
| have hη : ∀ᵐ x ∂η, dist x p ≤ 3 * R := by |
| change ∀ᵐ x ∂(q.map (fun γ => γ midpointTime) : Measure X), dist x p ≤ 3 * R |
| rw [ProbabilityMeasure.toMeasure_map] |
| apply (ae_map_iff (measurable_path_eval midpointTime).aemeasurable |
| (measurableSet_le (measurable_id.dist measurable_const) measurable_const)).mpr |
| exact hq.mono fun _ hγ => hγ.2 midpointTime |
| have hs0' : ∀ᵐ x ∂(ν₀ : Measure X), dist x p ≤ 3 * R := |
| hs0.mono fun _ hx => by linarith |
| obtain ⟨π, hπ, hg, hp, hi, hcomp⟩ := hX.exists_bounded_entropy_plan K n hn |
| (ν₀ : Measure X) η hA hb0 hb1 p (by positivity : 0 ≤ 3 * R) hs0' hη |
| let := hπ |
| have hqD : Integrable (fun γ : C(Time, X) => dist (γ 0) (γ 1) ^ 2) |
| (q : Measure C(Time, X)) := by |
| apply Integrable.mono' (integrable_const ((6 * R) ^ 2)) |
| (((measurable_path_eval 0).dist (measurable_path_eval 1)).pow_const 2).aestronglyMeasurable |
| filter_upwards [hq] with γ hγ |
| rw [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg _)] |
| have htri := dist_triangle (γ 0) p (γ 1) |
| rw [dist_comm p (γ 1)] at htri |
| exact pow_le_pow_left₀ dist_nonneg (by linarith [hγ.2 0, hγ.2 1]) 2 |
| have hqfinite : transportCost ((q : Measure C(Time, X)).map (fun γ => (γ 0, γ 1))) ≠ ∞ := by |
| rw [transportCost_plan_eq_integral _ hqD] |
| exact ENNReal.ofReal_ne_top |
| have hq1' : (q : Measure C(Time, X)).map (fun γ => γ 1) = Measure.dirac p := by |
| have hh := congrArg (fun ν : ProbabilityMeasure X => (ν : Measure X)) hq1 |
| change (q.map (fun γ => γ 1) : Measure X) = Measure.dirac p at hh |
| simpa only [ProbabilityMeasure.toMeasure_map] using hh |
| have hq0' : (q : Measure C(Time, X)).map (fun γ => γ 0) = (ν₀ : Measure X) := by |
| have hh := congrArg (fun ν : ProbabilityMeasure X => (ν : Measure X)) hq0 |
| simpa only [ProbabilityMeasure.toMeasure_map] using hh |
| have hp' : IsOptimalCoupling ((q : Measure C(Time, X)).map (fun γ => γ 0)) |
| ((q : Measure C(Time, X)).map (fun γ => γ midpointTime)) |
| (π.map (fun γ => (γ 0, γ 1))) := by |
| rw [hq0'] |
| exact hp |
| refine ⟨π, hπ, compressionBudget K n (2 * (3 * R)) A, |
| compressionBudget_ne_top K n _ hA, (optimalPlan_endpoints _ _ π hp).1, hg, |
| (memLp_two_iff_integrable_sq |
| ((measurable_path_eval 0).dist (measurable_path_eval 1)).aestronglyMeasurable).mpr hi, |
| fun t => (hcomp t).1, ?_⟩ |
| exact optimal_radial_midpoint q p (hq.mono fun _ hγ => hγ.1) hq1' hqfinite π hp' |
|
|
| end RegularPointCoordinates |
|
|
| end OAI |
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|
