| import OAI.Dynamics.SmoothObstruction.DensityBounds |
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| namespace OAI |
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| section |
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| namespace SmoothObstruction.HalfspaceCollar |
| open Set MeasureTheory Metric |
| open scoped ENNReal Manifold |
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| / |
| the original half-space. No extension across the boundary is used. -/ |
| theorem closedBall_interior {d : ℕ} [NeZero d] (x : EuclideanSpace ℝ (Fin d)) |
| {δ : ℝ} (hx : δ<x 0) : closedBall x δ⊆interior (Set.range (𝓡∂ d)) := by |
| intro y hy |
| rw [interior_range_modelWithCornersEuclideanHalfSpace] |
| have hh := PiLp.norm_apply_le (y-x) 0 |
| have hd : ‖y-x‖≤δ := by simpa only [mem_closedBall,dist_eq_norm] using hy |
| change 0<y 0 |
| simp only [PiLp.sub_apply,Real.norm_eq_abs] at hh |
| have h := (abs_le.mp (hh.trans hd)).1 |
| linarith |
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| / |
| knowing that the boundary is null would not imply the simultaneous |
| `[-n,n]` retained-mass estimate required in smooth switching. -/ |
| theorem slab_volume_le {d : ℕ} (j : Fin d) (R ε : ℝ) : |
| volume {x : EuclideanSpace ℝ (Fin d) | ‖x‖≤R ∧ 0≤x j ∧ x j≤ε}≤ |
| ENNReal.ofReal ε*(ENNReal.ofReal (2*R))^(d-1) := by |
| classical |
| let a : Fin d → ℝ := fun i => if i=j then 0 else -R |
| let b : Fin d → ℝ := fun i => if i=j then ε else R |
| have hsub : {x : EuclideanSpace ℝ (Fin d) | ‖x‖≤R ∧ 0≤x j ∧ x j≤ε}⊆ |
| WithLp.ofLp ⁻¹' Icc a b := by |
| intro x hx |
| constructor <;> intro i |
| · by_cases hi : i=j |
| · simpa [a,hi] using hx.2.1 |
| · have hh := (abs_le.mp ((PiLp.norm_apply_le x i).trans hx.1)).1 |
| simpa only [a,ite_eq_right hi] using hh |
| · by_cases hi : i=j |
| · simpa [b,hi] using hx.2.2 |
| · have hh := (abs_le.mp ((PiLp.norm_apply_le x i).trans hx.1)).2 |
| simpa only [b,ite_eq_right hi] using hh |
| calc |
| _ ≤ volume (WithLp.ofLp ⁻¹' Icc a b) := measure_mono hsub |
| _ = ∏ i,ENNReal.ofReal (b i-a i) := by |
| rw [(PiLp.volume_preserving_ofLp (Fin d)).measure_preimage measurableSet_Icc.nullMeasurableSet, |
| Real.volume_Icc_pi] |
| _ = ENNReal.ofReal ε*(ENNReal.ofReal (2*R))^(d-1) := by |
| rw [←Finset.mul_prod_erase _ _ (Finset.mem_univ j)] |
| have he : ∀ i∈Finset.univ.erase j,ENNReal.ofReal (b i-a i)=ENNReal.ofReal (2*R) := by |
| intro i hi |
| have hj : i≠j := (Finset.mem_erase.mp hi).1 |
| simp only [a,b,ite_eq_right hj,sub_neg_eq_add,←two_mul] |
| rw [Finset.prod_congr rfl he] |
| simp [a,b,Finset.card_erase_of_mem] |
|
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| end SmoothObstruction.HalfspaceCollar |
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|
| namespace SmoothObstruction.HalfspaceCollar |
| open Set MeasureTheory Metric |
| open scoped ENNReal |
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| / |
| the linear collar bound; the density is neither renormalized nor replaced |
| by a uniform conditional law. -/ |
| theorem supported_density_slab {d : ℕ} (j : Fin d) (R ε : ℝ) |
| (f : EuclideanSpace ℝ (Fin d) → ℝ≥0∞) (A : ℝ≥0∞) |
| (hf : ∀ x,f x≤A) (hzero : ∀ x,R<‖x‖ → f x=0) : |
| (volume.withDensity f) {x | 0≤x j ∧ x j≤ε}≤ |
| A*(ENNReal.ofReal ε*(ENNReal.ofReal (2*R))^(d-1)) := by |
| classical |
| have hdom : volume.withDensity f≤ |
| volume.withDensity ((closedBall (0 : EuclideanSpace ℝ (Fin d)) R).indicator (fun _ => A)) := by |
| apply withDensity_mono |
| apply Filter.Eventually.of_forall |
| intro x |
| by_cases hx : x∈closedBall (0 : EuclideanSpace ℝ (Fin d)) R |
| · simpa only [indicator_of_mem hx] using hf x |
| · rw [indicator_of_notMem hx,hzero x (by simpa only [mem_closedBall_zero_iff,not_le] using hx)] |
| rw [withDensity_indicator isClosed_closedBall.measurableSet,withDensity_const] at hdom |
| have hs : MeasurableSet {x : EuclideanSpace ℝ (Fin d) | 0≤x j ∧ x j≤ε} := by |
| exact (measurableSet_le measurable_const (show Measurable (fun x : EuclideanSpace ℝ (Fin d) => x j) by fun_prop)).inter |
| (measurableSet_le (show Measurable (fun x : EuclideanSpace ℝ (Fin d) => x j) by fun_prop) measurable_const) |
| calc |
| _ ≤ (A • volume.restrict (closedBall (0 : EuclideanSpace ℝ (Fin d)) R)) {x | 0≤x j ∧ x j≤ε} := |
| hdom _ |
| _ = A*volume {x : EuclideanSpace ℝ (Fin d) | ‖x‖≤R ∧ 0≤x j ∧ x j≤ε} := by |
| rw [Measure.smul_apply,Measure.restrict_apply hs] |
| change A * volume (_ ∩ _) = A * volume _ |
| congr 1 |
| congr 1 |
| ext x |
| simp only [mem_inter_iff,mem_ofPred_eq,mem_closedBall_zero_iff] |
| tauto |
| _ ≤ _ := by gcongr; exact slab_volume_le j R ε |
| end SmoothObstruction.HalfspaceCollar |
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|
| namespace SmoothObstruction.ChartDensity |
| open Set MeasureTheory Metric |
| open scoped Manifold ContDiff ENNReal |
| variable {d : ℕ} {H : Type*} [TopologicalSpace H] |
| (I : ModelWithCorners ℝ (EuclideanSpace ℝ (Fin d)) H) |
| {M : Type*} [TopologicalSpace M] [T2Space M] [ChartedSpace H M] |
| [MeasurableSpace M] [BorelSpace M] |
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| / |
| The measure is the original smooth measure restricted to that patch. -/ |
| theorem compact_chart_slab (hclosed : IsClosed (range I)) (μ : Measure M) |
| (hμ : SmoothPositiveDensity I μ) (a : M) (K : Set M) (hK : IsCompact K) |
| (hKU : K⊆(extChartAt I a).source) (j : Fin d) : |
| ∃ R A : ℝ,0<R ∧ 0<A ∧ ∀ ε : ℝ, |
| μ {x | x∈K ∧ 0≤coordinates I a x j ∧ coordinates I a x j≤ε}≤ |
| ENNReal.ofReal A*(ENNReal.ofReal ε*(ENNReal.ofReal (2*R))^(d-1)) := by |
| classical |
| obtain ⟨f,hf,_,hcell⟩ := cell_density I hclosed μ hμ a |
| have hKm : MeasurableSet K := hK.measurableSet |
| obtain ⟨_,hmap⟩ := hcell K hKm hKU |
| let e := coordinates I a |
| let W := ChartDomination.coordinateSet e (inverse I a) K |
| have him : W=(extChartAt I a) '' K := by |
| change ChartDomination.coordinateSet e (inverse I a) K=_ |
| rw [ChartDomination.coordinateSet_eq_image e (inverse I a) K |
| (fun x hx => inverse_coordinates I a x (hKU hx))] |
| apply image_congr |
| intro x hx |
| exact coordinates_eq I a x (hKU hx) |
| have hW : IsCompact W := by |
| rw [him] |
| exact hK.image_of_continuousOn ((continuousOn_extChartAt a).mono hKU) |
| have hWV : W⊆(extChartAt I a).target := by |
| rw [him] |
| rintro u ⟨x,hx,rfl⟩ |
| exact (extChartAt I a).map_source (hKU hx) |
| obtain ⟨R₀,hR₀⟩ := hW.bddAbove_image continuous_norm.continuousOn |
| obtain ⟨A₀,hA₀⟩ := hW.bddAbove_image (hf.continuousOn.mono hWV) |
| let R := max R₀ 0+1 |
| let A := max A₀ 0+1 |
| have hR : 0<R := by dsimp [R]; positivity |
| have hA : 0<A := by dsimp [A]; positivity |
| have hnorm (u) (hu : u∈W) : ‖u‖≤R := by |
| have hh := hR₀ (mem_image_of_mem (fun u => ‖u‖) hu) |
| exact hh.trans ((le_max_left _ _).trans (le_add_of_nonneg_right zero_le_one)) |
| have hbound (u) (hu : u∈W) : f u≤A := by |
| have hh := hA₀ (mem_image_of_mem f hu) |
| exact hh.trans ((le_max_left _ _).trans (le_add_of_nonneg_right zero_le_one)) |
| refine ⟨R,A,hR,hA,fun ε => ?_⟩ |
| let V : Set (EuclideanSpace ℝ (Fin d)) := {u | 0≤u j ∧ u j≤ε} |
| have hV : MeasurableSet V := |
| (measurableSet_le measurable_const (show Measurable (fun u : EuclideanSpace ℝ (Fin d) => u j) by fun_prop)).inter |
| (measurableSet_le (show Measurable (fun u : EuclideanSpace ℝ (Fin d) => u j) by fun_prop) measurable_const) |
| have he : Measurable e := measurable_coordinates I a |
| have hid : μ {x | x∈K ∧ 0≤e x j ∧ e x j≤ε} = |
| ((μ.restrict K).map e) V := by |
| rw [Measure.map_apply he hV,Measure.restrict_apply (hV.preimage he)] |
| congr 1 |
| ext x |
| simp only [mem_ofPred_eq,mem_inter_iff,mem_preimage,V] |
| tauto |
| rw [hid,hmap] |
| apply HalfspaceCollar.supported_density_slab j R ε |
| · intro u |
| change W.indicator (fun u => ENNReal.ofReal (f u)) u≤ENNReal.ofReal A |
| by_cases hu : u∈W |
| · simpa only [indicator_of_mem hu] using ENNReal.ofReal_le_ofReal (hbound u hu) |
| · simp only [indicator_of_notMem hu,zero_le] |
| · intro u hu |
| exact indicator_of_notMem (fun hw => (not_lt_of_ge (hnorm u hw)) hu) _ |
| end SmoothObstruction.ChartDensity |
|
|
| namespace SmoothObstruction.ChartDensity |
| open Set MeasureTheory Metric |
| open scoped Manifold ContDiff ENNReal |
| variable {d : ℕ} [NeZero d] |
| {M : Type*} [TopologicalSpace M] [T2Space M] [ChartedSpace (EuclideanHalfSpace d) M] |
| [MeasurableSpace M] [BorelSpace M] |
| {ι : Type*} [Fintype ι] |
|
|
| / |
| positive smooth density and authentic finite compact atlas. The same set |
| works for every selected chart containing the point; its mass is O(δ), not |
| merely o(1), so all 2n+1 tests can be charged to invariant volume. -/ |
| theorem global_halfspace_collar (μ : Measure M) [IsFiniteMeasure μ] |
| (hμ : SmoothPositiveDensity (𝓡∂ d) μ) (a : ι → M) (K : ι → Set M) |
| (hK : ∀ i,IsCompact (K i)) (hKU : ∀ i,K i⊆(chartAt (EuclideanHalfSpace d) (a i)).source) : |
| ∃ C : ℝ,0<C ∧ ∀ δ : ℝ,0≤δ → |
| ∃ W : Set M,MeasurableSet W ∧ μ.real W≤C*δ ∧ |
| ∀ i x,x∈K i → x∉W → |
| closedBall ((extChartAt (𝓡∂ d) (a i)) x) δ⊆interior (range (𝓡∂ d)) := by |
| classical |
| have hclosed : IsClosed (range (𝓡∂ d)) := by |
| rw [range_modelWithCornersEuclideanHalfSpace] |
| exact isClosed_le continuous_const (by fun_prop) |
| have hs (i : ι) := compact_chart_slab (𝓡∂ d) hclosed μ hμ (a i) (K i) (hK i) |
| (by simpa only [extChartAt_source] using hKU i) (0 : Fin d) |
| choose R A hR hA hm using hs |
| let D (i : ι) := A i*(2*R i)^(d-1) |
| let C := (∑ i,D i)+1 |
| have hD (i : ι) : 0≤D i := by exact mul_nonneg (hA i).le (pow_nonneg (by linarith [hR i]) _) |
| have hC : 0<C := by have := Finset.sum_nonneg (fun i (_ : i∈Finset.univ) => hD i); dsimp [C]; linarith |
| refine ⟨C,hC,?_⟩ |
| intro δ hδ |
| let V (i : ι) : Set M := {x | x∈K i ∧ 0≤coordinates (𝓡∂ d) (a i) x 0 ∧ |
| coordinates (𝓡∂ d) (a i) x 0≤δ} |
| let W := ⋃ i,V i |
| have hV (i : ι) : MeasurableSet (V i) := by |
| have hf : Measurable (fun x => coordinates (𝓡∂ d) (a i) x 0) := by |
| exact (show Measurable (fun u : EuclideanSpace ℝ (Fin d) => u 0) by fun_prop).comp (measurable_coordinates (𝓡∂ d) (a i)) |
| exact (hK i).measurableSet.inter ((measurableSet_le measurable_const hf).inter |
| (measurableSet_le hf measurable_const)) |
| have hVm (i : ι) : μ (V i)≤ENNReal.ofReal (D i*δ) := by |
| have hh := hm i δ |
| have he : ENNReal.ofReal (A i)*(ENNReal.ofReal δ*(ENNReal.ofReal (2*R i))^(d-1))= |
| ENNReal.ofReal (D i*δ) := by |
| rw [ENNReal.ofReal_mul (hD i),ENNReal.ofReal_mul (hA i).le,ENNReal.ofReal_pow (by linarith [hR i])] |
| ring |
| exact hh.trans_eq he |
| have hWm : μ W≤ENNReal.ofReal (C*δ) := by |
| calc |
| μ W ≤ ∑ i,μ (V i) := by simpa only [tsum_fintype] using measure_iUnion_le (μ:=μ) V |
| _ ≤ ∑ i,ENNReal.ofReal (D i*δ) := Finset.sum_le_sum (fun i _ => hVm i) |
| _ = ENNReal.ofReal ((∑ i,D i)*δ) := by |
| rw [Finset.sum_mul,ENNReal.ofReal_sum_of_nonneg (fun i _ => mul_nonneg (hD i) hδ)] |
| _ ≤ ENNReal.ofReal (C*δ) := ENNReal.ofReal_le_ofReal |
| (mul_le_mul_of_nonneg_right (by dsimp [C]; linarith) hδ) |
| refine ⟨W,MeasurableSet.iUnion hV,?_,?_⟩ |
| · exact (ENNReal.toReal_mono ENNReal.ofReal_ne_top hWm).trans_eq (ENNReal.toReal_ofReal (mul_nonneg hC.le hδ)) |
| · intro i x hx hxW |
| have hxs : x∈(extChartAt (𝓡∂ d) (a i)).source := by simpa only [extChartAt_source] using hKU i hx |
| have hxrange := (extChartAt (𝓡∂ d) (a i)).map_source hxs |
| rw [extChartAt,OpenPartialHomeomorph.extend_target] at hxrange |
| have hxnonneg : 0≤(extChartAt (𝓡∂ d) (a i) x) 0 := by |
| have hh := hxrange.2 |
| rw [range_modelWithCornersEuclideanHalfSpace] at hh |
| exact hh |
| have hxnot : x∉V i := fun hh => hxW (mem_iUnion.mpr ⟨i,hh⟩) |
| apply HalfspaceCollar.closedBall_interior |
| by_contra hle |
| apply hxnot |
| refine ⟨hx,?_,?_⟩ <;> rw [coordinates_eq (𝓡∂ d) _ _ hxs] |
| · exact hxnonneg |
| · exact le_of_not_gt hle |
| end SmoothObstruction.ChartDensity |
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| end |
|
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| end OAI |
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