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e4d9d2b | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 | 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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