| import Mathlib |
| import OAI.Analysis.BiholderTransport.Geodesics.SplitMinimizingAction |
| import OAI.Analysis.BiholderTransport.CostGeometry.CostSmooth |
| import OAI.Analysis.BiholderTransport.Coordinates.ManifoldMinimum |
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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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| open Set Filter |
| open scoped Topology ContDiff |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
|
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
|
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
|
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set Filter |
| open scoped ContDiff Topology |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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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] |
|
|
| lemma cost_start_gradient_of_injectivityDomain {x : M} {p : TangentSpace 𝓘(ℝ,Model n) x} |
| (hp : p ∈ injectivityDomain x) : |
| HasMFDerivAt 𝓘(ℝ,Model n) 𝓘(ℝ,ℝ) (fun y => cost y (riemannianExp x p)) |
| x (innerSL ℝ (-p)) := by |
| have hpmin := injectivityDomain_subset_minimizingVectors x hp |
| obtain ⟨δ,hδ,H⟩ := exists_local_spray_cost_start_gradient (E := Model n) x |
| have hnear : ∀ᶠ s : ℝ in 𝓝 0, riemannianExp x (s • p) ∈ Metric.ball x δ := by |
| have ht : ContinuousAt (fun s : ℝ => riemannianExp x (s • p)) 0 := |
| ((continuous_riemannianExp x).comp (continuous_id.smul continuous_const)).continuousAt |
| have hx : riemannianExp x ((0:ℝ) • p) ∈ Metric.ball x δ := by simpa using hδ |
| exact ht.preimage_mem_nhds (Metric.isOpen_ball.mem_nhds hx) |
| have hsmall : ∀ᶠ s : ℝ in 𝓝[>] 0, |
| riemannianExp x (s • p) ∈ Metric.ball x δ ∧ s<1 ∧ 0<s := |
| (hnear.filter_mono nhdsWithin_le_nhds).and |
| (((show ∀ᶠ r : ℝ in 𝓝 0, r<1 from eventually_lt_nhds (show (0:ℝ)<1 by norm_num)).filter_mono nhdsWithin_le_nhds).and self_mem_nhdsWithin) |
| obtain ⟨s,hy,hs1,hs⟩ := hsmall.exists |
| have hz : x ∈ Metric.ball x δ := by simpa using hδ |
| have hys : (sprayFlow s (⟨x,p⟩ : TangentBundle 𝓘(ℝ,Model n) M)).1 ∈ Metric.ball x δ := by |
| rwa [←riemannianExp_smul] |
| have hmin : dist x (sprayFlow s (⟨x,p⟩ : TangentBundle 𝓘(ℝ,Model n) M)).1=s*‖p‖ := by |
| rw [←riemannianExp_smul,minimizingVectors_smul hpmin hs.le hs1.le,norm_smul,Real.norm_eq_abs,abs_of_pos hs] |
| have hD := H ⟨x,p⟩ s hs hz hys hmin |
| rw [←riemannianExp_smul] at hD |
| dsimp only at hD |
| let G : M → ℝ := fun y => cost y (riemannianExp x (s • p))/s+(1-s)*‖p‖^2/2 |
| have hG : HasMFDerivAt 𝓘(ℝ,Model n) 𝓘(ℝ,ℝ) G x (innerSL ℝ (-p)) := by |
| have HH := (hD.const_smul (s⁻¹)).add |
| (hasMFDerivAt_const (I := 𝓘(ℝ,Model n)) (I' := 𝓘(ℝ,ℝ)) ((1-s)*‖p‖^2/2) x) |
| have hGG : G =ᶠ[𝓝 x] |
| ((s⁻¹ • fun y => dist y (riemannianExp x (s • p))^2/2) + fun _ => (1-s)*‖p‖^2/2) := by |
| apply Filter.Eventually.of_forall |
| intro y |
| dsimp [G,cost] |
| simp only [div_eq_mul_inv] |
| ring |
| apply (HH.congr_of_eventuallyEq hGG).congr_mfderiv |
| have heq : (s⁻¹ • innerSL ℝ ((-s) • p)) + 0 = innerSL ℝ (-p) := by |
| ext v |
| simp only [_root_.smul_apply,add_zero,innerSL_apply_apply,real_inner_smul_left,inner_neg_left,smul_eq_mul] |
| field_simp [ne_of_gt hs] |
| convert! heq using 1 |
| have hcost := cost_contMDiffAt_of_injectivityDomain (⟨x,p⟩ : TangentBundle 𝓘(ℝ,Model n) M) hp |
| have hF : MDifferentiableAt 𝓘(ℝ,Model n) 𝓘(ℝ,ℝ) |
| (fun y => cost y (riemannianExp x p)) x := |
| ((hcost.comp x (contMDiffAt_id.prodMk contMDiffAt_const)).mdifferentiableAt (by simp)) |
| have heq := upper_contact_mfderiv hF hG.mdifferentiableAt |
| (Filter.Eventually.of_forall (minimizing_split_upper_support hpmin hs hs1)) |
| (minimizing_split_contact hpmin hs hs1).symm |
| exact hF.hasMFDerivAt.congr_mfderiv (heq.trans hG.mfderiv) |
| |
| end WeakMTWTransport |
| |
| end |
| |
| end OAI |
| |
