File size: 1,254 Bytes
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 | 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
|