| import Mathlib |
| import OAI.Analysis.BiholderTransport.LinearAlgebra.CurvedBlowup |
| import OAI.Analysis.BiholderTransport.CostGeometry.FirstCutLower |
| import OAI.Analysis.BiholderTransport.Coordinates.KernelRadial |
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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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| 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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| 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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| 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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| 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 Filter Set |
| open scoped 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 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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| 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 kernel_parallel_of_transverse_orthogonal_ne_zero |
| {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
| {e k : E} (he : e ≠ 0) |
| (hk : ∀ xi : E, inner ℝ xi e=0 → inner ℝ xi k=0) : |
| k=(inner ℝ k e / inner ℝ e e) • e := by |
| have hee : inner ℝ e e ≠ 0 := inner_self_ne_zero.mpr he |
| let xi := k-(inner ℝ k e / inner ℝ e e) • e |
| have hxi : inner ℝ xi e=0 := by |
| dsimp [xi] |
| simp only [inner_sub_left,real_inner_smul_left,div_mul_cancel₀ _ hee,sub_self] |
| have hxik := hk xi hxi |
| have hxixi : inner ℝ xi xi=0 := by |
| change inner ℝ xi (k-(inner ℝ k e / inner ℝ e e) • e)=0 |
| simp only [inner_sub_right,real_inner_smul_right,hxik,hxi,mul_zero,sub_zero] |
| exact sub_eq_zero.mp (inner_self_eq_zero.mp hxixi) |
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| lemma WeakMTW.firstcut_segment_kernel_zero (hmtw : WeakMTW (n := n) (M := M)) |
| {x : M} {p e k : TangentSpace 𝓘(ℝ,Model n) x} {a b : ℝ} |
| (ha : a<0) (hb : 0<b) (he : e≠0) |
| (hseg : ∀ s ∈ Icc a b, p+s • e ∈ minimizingVectors x) |
| (hleft : Function.Injective (fderiv ℝ (fun v => extChartAt 𝓘(ℝ,Model n) |
| (riemannianExp x (p+a • e)) (riemannianExp x v)) (p+a • e))) |
| (hright : Function.Injective (fderiv ℝ (fun v => extChartAt 𝓘(ℝ,Model n) |
| (riemannianExp x (p+b • e)) (riemannianExp x v)) (p+b • e))) |
| (hk : mfderiv 𝓘(ℝ,TangentSpace 𝓘(ℝ,Model n) x) 𝓘(ℝ,Model n) (riemannianExp x) p k=0) : |
| k=0 := by |
| by_contra hk0 |
| have hpar := kernel_parallel_of_transverse_orthogonal_ne_zero he |
| (hmtw.firstcut_kernel_transverse_orthogonal ha hb hseg hleft hright hk) |
| have hpair : inner ℝ k e≠0 := by |
| intro hh |
| rw [hh,zero_div,zero_smul] at hpar |
| exact hk0 hpar |
| have hortho : inner ℝ p e=0 := by |
| have hh := exp_kernel_radial_orthogonal hk |
| rw [hpar,real_inner_smul_right] at hh |
| exact (mul_eq_zero.mp hh).resolve_left (div_ne_zero hpair (inner_self_ne_zero.mpr he)) |
| have hp : p ∈ minimizingVectors x := by simpa using hseg 0 ⟨ha.le,hb.le⟩ |
| have hnear : ∀ᶠ s : ℝ in 𝓝 0, p+s • e ∈ minimizingVectors x := by |
| filter_upwards [Ioo_mem_nhds ha hb] with s hs |
| exact hseg s ⟨hs.1.le,hs.2.le⟩ |
| obtain ⟨eps,B,c,heps,hc,hupper⟩ := conjugate_curved_hessian_upper hp hnear hk hpair |
| obtain ⟨D,hD⟩ := hmtw.firstcut_radial_lower hb |
| (fun s hs => hseg s ⟨ha.le.trans hs.1,hs.2⟩) hright hortho |
| have hcurve : Tendsto (fun s : ℝ => 1-s^2) (𝓝[>] 0) (𝓝[<] 1) := by |
| apply tendsto_nhdsWithin_iff.mpr |
| constructor |
| · simpa using (show ContinuousAt (fun s : ℝ => 1-s^2) 0 from by fun_prop).tendsto.mono_left nhdsWithin_le_nhds |
| · filter_upwards [self_mem_nhdsWithin] with s hs |
| change 1-s^2<1 |
| have hs0 : 0<s := hs |
| nlinarith [sq_pos_of_pos hs0] |
| have hpoly : ∀ᶠ s : ℝ in 𝓝[>] 0, c≤D*s+B*s^2 := by |
| filter_upwards [hcurve.eventually hD,self_mem_nhdsWithin, |
| (eventually_lt_nhds heps).filter_mono nhdsWithin_le_nhds, |
| (eventually_lt_nhds hb).filter_mono nhdsWithin_le_nhds] with s hs hs0 hseps hsb |
| have hspos : 0<s := hs0 |
| have hlo := hs s ⟨hspos.le,hsb.le⟩ |
| have hhi := hupper s ⟨hspos,hseps⟩ |
| have hdiv : c/s^2≤B-hessianValue x ((1-s^2) • (p+s • e)) e := by linarith |
| have hmul := (div_le_iff₀ (sq_pos_of_pos hspos)).mp hdiv |
| nlinarith |
| have htend : Tendsto (fun s : ℝ => D*s+B*s^2) (𝓝[>] 0) (𝓝 0) := by |
| simpa using (show ContinuousAt (fun s : ℝ => D*s+B*s^2) 0 from by fun_prop).tendsto.mono_left nhdsWithin_le_nhds |
| have hc0 : c≤0 := le_of_tendsto_of_tendsto tendsto_const_nhds htend hpoly |
| exact (not_le_of_gt hc) hc0 |
| end WeakMTWTransport |
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| end |
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| end OAI |
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