| import Mathlib |
| import OAI.Analysis.BiholderTransport.Geodesics.MinimizingSubinterval |
| import OAI.Analysis.BiholderTransport.CostGeometry.DividedAction |
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|
| namespace OAI |
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| noncomputable section |
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| open Set MeasureTheory Manifold Bundle |
| open scoped ContDiff Manifold ENNReal NNReal Topology |
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| open Set Filter |
| open scoped Topology NNReal |
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| open Set Filter |
| open scoped Topology |
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| open Set Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set |
| open scoped Topology |
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| open Set Filter Manifold Bundle ContinuousLinearMap |
| open scoped Topology ContDiff Manifold Bundle |
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| open Set Filter ContinuousLinearMap InnerProductSpace |
| open scoped Topology ContDiff |
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| open Set Filter ContinuousLinearMap |
| open scoped Topology ContDiff |
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| open Set Filter ContinuousLinearMap |
| open scoped Topology ContDiff |
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| open Set Filter ContinuousLinearMap |
| open scoped Topology ContDiff |
| open scoped NNReal |
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| open Set Filter ContinuousLinearMap |
| open scoped Topology ContDiff |
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| open Set Filter ContinuousLinearMap |
| open scoped Topology |
| open MeasureTheory |
| open scoped ContDiff ENNReal |
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| open Set Filter Manifold Bundle ContinuousLinearMap MeasureTheory |
| open scoped Topology ContDiff Manifold Bundle ENNReal |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology Manifold |
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| open Set Filter Manifold Bundle ContinuousLinearMap |
| open scoped Topology ContDiff Manifold Bundle |
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| open Set Filter Manifold Bundle |
| open scoped Topology ContDiff Manifold Bundle |
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| open Set Filter Manifold Bundle |
| open scoped Topology ContDiff Manifold Bundle |
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| open Set Filter Bundle |
| open scoped Topology Bundle |
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| open scoped Topology |
| open Function Manifold Set |
| open Manifold Bundle |
| open scoped Manifold Bundle |
| open Set |
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| namespace WeakMTWTransport |
| variable {n : ℕ} {M : Type*} [MetricSpace M] [CompactSpace M] |
| [ChartedSpace (Model n) M] [IsManifold 𝓘(ℝ,Model n) ∞ M] |
| [RiemannianBundle (fun x : M => TangentSpace 𝓘(ℝ,Model n) x)] |
| [IsContMDiffRiemannianBundle 𝓘(ℝ,Model n) ∞ (Model n) |
| (fun x : M => TangentSpace 𝓘(ℝ,Model n) x)] |
| [IsRiemannianManifold 𝓘(ℝ,Model n) M] |
|
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| lemma minimizing_split_upper_support {x : M} {p : TangentSpace 𝓘(ℝ,Model n) x} |
| (hp : p ∈ minimizingVectors x) {s : ℝ} (hs : 0<s) (hs1 : s<1) (z : M) : |
| cost z (riemannianExp x p) ≤ cost z (riemannianExp x (s • p))/s + (1-s)*‖p‖^2/2 := by |
| have hseg : dist (riemannianExp x (s • p)) (riemannianExp x p) = (1-s)*‖p‖ := by |
| rw [riemannianExp_smul,riemannianExp_eq_sprayFlow] |
| have hh := sprayFlow_minimizing_subinterval (⟨x,p⟩ : TangentBundle 𝓘(ℝ,Model n) M) |
| hs.le hs1.le (show dist x (sprayFlow 1 ⟨x,p⟩).1 = 1*‖p‖ by rw [one_mul,←riemannianExp_eq_sprayFlow]; exact hp) |
| exact hh |
| have hh := squared_dist_divided_action_le z (riemannianExp x (s • p)) (riemannianExp x p) hs hs1 |
| rw [hseg] at hh |
| unfold cost |
| convert hh using 1 |
| field_simp [hs.ne', (sub_pos.mpr hs1).ne'] |
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| lemma minimizing_split_contact {x : M} {p : TangentSpace 𝓘(ℝ,Model n) x} |
| (hp : p ∈ minimizingVectors x) {s : ℝ} (hs : 0<s) (hs1 : s<1) : |
| cost x (riemannianExp x (s • p))/s + (1-s)*‖p‖^2/2 = cost x (riemannianExp x p) := by |
| have hsp := minimizingVectors_smul hp hs.le hs1.le |
| unfold cost |
| rw [hsp,hp,norm_smul,Real.norm_eq_abs,abs_of_pos hs] |
| field_simp [hs.ne'] |
| ring |
| |
| end WeakMTWTransport |
| |
| end |
| |
| end OAI |
| |
