| import OAI.Geometry.RCD.Coordinates.NormalizedSubplanNodes |
|
|
| namespace OAI |
|
|
| /-! # Half nodes for an actual compressed plan -/ |
|
|
| open Set Filter MeasureTheory Metric TopologicalSpace |
| open scoped ENNReal Topology |
| noncomputable section |
| namespace RegularPointCoordinates |
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] |
| [SeparableSpace X] [CompleteSpace X] |
| attribute [local instance] pathBorel pathBorelSpace |
|
|
| theorem compressed_plan_children {n : ℕ} (a : RenyiTransportNode n X) |
| (π : Measure C(Time,X)) [IsProbabilityMeasure π] |
| (hg : ∀ᵐ γ ∂π, IsGeodesic γ) |
| (hp : IsOptimalCoupling a.left a.right (π.map (fun γ => (γ 0,γ 1)))) |
| {C : ℝ≥0∞} (hC : C≠∞) (hc : ∀ t : Time, π.map (fun γ => γ t)≤C • hausdorff n X) : |
| ∃ a₀ a₁ : RenyiTransportNode n X, |
| a₀.left=π.map (fun γ => γ 0) ∧ a₀.right=π.map (fun γ => γ midpointTime) ∧ |
| a₁.left=π.map (fun γ => γ midpointTime) ∧ a₁.right=π.map (fun γ => γ 1) ∧ |
| a₀.distance=a.distance/2 ∧ a₁.distance=a.distance/2 := by |
| simpa only [measure_univ,inv_one,one_smul] using |
| exists_normalized_subplan_half_nodes a π hg hp hC hc π le_rfl (by simp) |
|
|
| end RegularPointCoordinates |
|
|
| end |
|
|
| end OAI |
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|
