| import Mathlib |
| import OAI.Probability.LatticeSurface.Convexity.RadialExposure |
| import OAI.Probability.LatticeSurface.Geometry.BlockFacetGeometry |
| import OAI.Probability.LatticeSurface.Geometry.BlockVolume |
| import OAI.Probability.LatticeSurface.Geometry.EdgeCoordinates |
| import OAI.Probability.LatticeSurface.Geometry.FacetExposure |
| import OAI.Probability.LatticeSurface.Geometry.TriangularCover |
| import OAI.Probability.LatticeSurface.Geometry.TriangularGeometry |
|
|
| namespace OAI |
|
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| open MeasureTheory |
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| open scoped ENNReal |
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| noncomputable section |
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| namespace TriangularLipschitz |
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| end TriangularLipschitz |
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| namespace FacetExposure |
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| end FacetExposure |
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| namespace TriangularLipschitz |
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| end TriangularLipschitz |
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| namespace TriangularLipschitz |
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| end TriangularLipschitz |
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| namespace TriangularLipschitz |
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| end TriangularLipschitz |
|
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| open MeasureTheory Set |
|
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| open scoped ENNReal |
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| noncomputable section |
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| namespace EdgeCoordinates |
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| end EdgeCoordinates |
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| namespace EdgeCoordinates |
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| end EdgeCoordinates |
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| namespace EdgeCoordinates |
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| end EdgeCoordinates |
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| namespace TriangularLipschitz |
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|
| noncomputable def edgeOffset (L : ℕ) (τ : ℝ) (s : Vertex → ℝ) (x : Vertex) : ℝ := |
| if x ∈ interior L then 0 else τ*s x |
|
|
| lemma exists_filling_edge_coordinates {L n : ℕ} |
| (e : {x : Vertex |
| {x y : Vertex} (hy : y ∈ interior L) (hxy : x ≠ y) {τ : ℝ} (hτ : |τ|=1) : |
| ∃ E : Filling L ≃ₗ[ℝ] (Fin (n+1) → ℝ), |
| MeasurePreserving E ∧ (∀ z, ‖E.symm z‖ ≤ 2*‖z‖) ∧ |
| ∀ s a, E a 0 = τ*(filledHeight L s a y-filledHeight L s a x)+edgeOffset L τ s x := by |
| classical |
| by_cases hx : x ∈ interior L |
| · obtain ⟨E,hE,hEn,hE0⟩ := EdgeCoordinates.exists_edge_coordinates e ⟨y,hy⟩ |
| (some ⟨x,hx⟩) τ hτ (by intro i hi he; cases hi; exact hxy (congrArg Subtype.val he)) |
| refine ⟨E,hE,hEn,?_⟩ |
| intro s a |
| simpa [filledHeight,edgeOffset,hx,hy] using hE0 a |
| · obtain ⟨E,hE,hEn,hE0⟩ := EdgeCoordinates.exists_edge_coordinates e ⟨y,hy⟩ |
| none τ hτ (by simp) |
| refine ⟨E,hE,hEn,?_⟩ |
| intro s a |
| simp only [hE0,Option.elim_none,sub_zero,filledHeight,dite_eq_left hy,dite_eq_right hx,edgeOffset,ite_eq_right hx] |
| ring |
|
|
| lemma filling_projection_width {L n : ℕ} {x y : Vertex} {τ : ℝ} (s : Vertex → ℝ) |
| (hτ : |τ|=1) (hx : x ∈ box L) (hy : y ∈ box L) (hxy : Adj x y) |
| (E : Filling L ≃ₗ[ℝ] (Fin (n+1) → ℝ)) |
| (hE0 : ∀ s a, E a 0 = τ*(filledHeight L s a y-filledHeight L s a x)+edgeOffset L τ s x) : |
| ∀ z ∈ E '' fillingPolytope L s, |
| z 0 ∈ Set.Icc (1+edgeOffset L τ s x-2) (1+edgeOffset L τ s x) := by |
| rintro z ⟨a,ha,rfl⟩ |
| rw [hE0] |
| have h := ha x hx y hy hxy |
| have h' : |τ*(filledHeight L s a y-filledHeight L s a x)| ≤ 1 := by |
| rw [abs_mul,hτ,one_mul,abs_sub_comm]; exact h |
| obtain ⟨h₁,h₂⟩ := abs_le.mp h' |
| exact ⟨by linarith,by linarith⟩ |
|
|
| lemma filling_facet_ball {L n : ℕ} {x y : Vertex} {s : Vertex → ℝ} {τ β : ℝ} |
| (hτ : |τ|=1) (hβ : 0 < β) |
| (E : Filling L ≃ₗ[ℝ] (Fin (n+1) → ℝ)) |
| (hEn : ∀ z, ‖E.symm z‖ ≤ 2*‖z‖) |
| (hE0 : ∀ s a, E a 0 = τ*(filledHeight L s a y-filledHeight L s a x)+edgeOffset L τ s x) |
| (a₀ : Filling L) |
| (hfacet : τ*(filledHeight L s a₀ y-filledHeight L s a₀ x)=1) |
| (hother : ∀ v ∈ box L, ∀ w ∈ box L, Adj v w → s(v,w) ≠ s(x,y) → |
| |filledHeight L s a₀ v-filledHeight L s a₀ w| ≤ 1-β/2) : |
| ∃ q₀ : Fin n → ℝ, ∀ q ∈ Metric.ball q₀ (β/16), |
| Fin.cons (1+edgeOffset L τ s x) q ∈ E '' fillingPolytope L s := by |
| let q₀ : Fin n → ℝ := fun i => E a₀ i.succ |
| refine ⟨q₀,?_⟩ |
| intro q hq |
| let z : Fin (n+1) → ℝ := Fin.cons (1+edgeOffset L τ s x) q |
| let a := E.symm z |
| have hb : E a₀ 0=1+edgeOffset L τ s x := by rw [hE0,hfacet] |
| have hnorm : ‖z-E a₀‖ ≤ ‖q-q₀‖ := by |
| apply (pi_norm_le_iff_of_nonneg (norm_nonneg _)).mpr |
| intro i |
| refine Fin.cases ?_ (fun j => ?_) i |
| · simp [z,hb,norm_nonneg] |
| · exact norm_le_pi_norm (q-q₀) j |
| have ha : ‖a-a₀‖ ≤ 2*‖q-q₀‖ := by |
| have heq : a-a₀=E.symm (z-E a₀) := by simp [a] |
| rw [heq] |
| exact (hEn _).trans (mul_le_mul_of_nonneg_left hnorm (by norm_num)) |
| have hq' : ‖q-q₀‖ < β/16 := by simpa only [Metric.mem_ball,dist_eq_norm] using hq |
| have hsign : τ*(filledHeight L s a y-filledHeight L s a x)=1 := by |
| have h := hE0 s a |
| have hz : E a 0=1+edgeOffset L τ s x := by simp [a,z] |
| linarith |
| have habs : |filledHeight L s a x-filledHeight L s a y| ≤ 1 := by |
| have h := congrArg abs hsign |
| rw [abs_mul,hτ,one_mul,abs_one,abs_sub_comm] at h |
| exact h.le |
| refine ⟨a,filling_of_small_perturbation_on_facet habs hother (by linarith),?_⟩ |
| exact E.apply_symm_apply z |
| |
| end TriangularLipschitz |
| |
| namespace TriangularLipschitz |
| |
| lemma linearCoordinates_measurableEmbedding {L n : ℕ} |
| (E : Filling L ≃ₗ[ℝ] (Fin n → ℝ)) : MeasurableEmbedding E := |
| E.toContinuousLinearEquiv.toHomeomorph.isClosedEmbedding.measurableEmbedding |
| |
| lemma filling_volume_image {L n : ℕ} (E : Filling L ≃ₗ[ℝ] (Fin n → ℝ)) |
| (hE : MeasurePreserving E) (s : Vertex → ℝ) : |
| volume (E '' fillingPolytope L s) = fillingVolume L s := by |
| have h := hE.measure_preimage_emb (linearCoordinates_measurableEmbedding E) (E '' fillingPolytope L s) |
| simpa only [Set.preimage_image_eq _ E.injective, fillingVolume] using h.symm |
| |
| lemma isCompact_filling_image {L n : ℕ} (E : Filling L ≃ₗ[ℝ] (Fin n → ℝ)) (s : Vertex → ℝ) : |
| IsCompact (E '' fillingPolytope L s) := |
| (isCompact_fillingPolytope L s).image E.toContinuousLinearEquiv.continuous |
| |
| lemma convex_filling_image {L n : ℕ} (E : Filling L ≃ₗ[ℝ] (Fin n → ℝ)) (s : Vertex → ℝ) : |
| Convex ℝ (E '' fillingPolytope L s) := |
| (convex_fillingPolytope L s).linear_image E.toLinearMap |
| |
| lemma filling_image_volume_bound {L n : ℕ} (E : Filling L ≃ₗ[ℝ] (Fin n → ℝ)) |
| (hE : MeasurePreserving E) (s : Vertex → ℝ) : |
| (volume (E '' fillingPolytope L s)).toReal ≤ |
| (volume (Metric.closedBall (0 : Filling L) (2*L))).toReal := by |
| rw [filling_volume_image E hE] |
| have h : (fillingVolume L s).toReal ≤ |
| (volume (Metric.closedBall (fun _ => s (0,0) : Filling L) (2*L))).toReal := |
| ENNReal.toReal_mono measure_closedBall_lt_top.ne |
| (measure_mono (fillingPolytope_subset_closedBall L s)) |
| simpa only [Measure.addHaar_closedBall_center] using h |
| |
| def meanHeight (L : ℕ) (s : Vertex → ℝ) (x : Vertex) : ℝ := |
| (∫ a in fillingPolytope L s, filledHeight L s a x)/(fillingVolume L s).toReal |
| |
| lemma integrableOn_filledHeight (L : ℕ) (s : Vertex → ℝ) (x : Vertex) : |
| IntegrableOn (fun a => filledHeight L s a x) (fillingPolytope L s) := |
| (continuous_filledHeight L s x).continuousOn.integrableOn_compact (isCompact_fillingPolytope L s) |
| |
| lemma meanHeight_fixed {L : ℕ} {s : Vertex → ℝ} {x : Vertex} |
| (hvol : 0 < fillingVolume L s) (hx : x ∉ interior L) : meanHeight L s x=s x := by |
| have hv : (fillingVolume L s).toReal ≠ 0 := |
| (ENNReal.toReal_pos hvol.ne' (fillingVolume_lt_top L s).ne).ne' |
| simp only [meanHeight,filledHeight_fixed _ _ _ hx, setIntegral_const,smul_eq_mul, |
| Measure.real, fillingVolume] at hv ⊢ |
| exact mul_div_cancel_left₀ _ hv |
| |
| lemma coordinate_meanHeight {L n : ℕ} {x y : Vertex} {τ : ℝ} |
| (E : Filling L ≃ₗ[ℝ] (Fin (n+1) → ℝ)) (hE : MeasurePreserving E) |
| (hE0 : ∀ s a, E a 0 = τ*(filledHeight L s a y-filledHeight L s a x)+edgeOffset L τ s x) |
| (s : Vertex → ℝ) (hvol : 0 < fillingVolume L s) : |
| (∫ z in E '' fillingPolytope L s, z 0)/(volume (E '' fillingPolytope L s)).toReal = |
| τ*(meanHeight L s y-meanHeight L s x)+edgeOffset L τ s x := by |
| have hv : (fillingVolume L s).toReal ≠ 0 := |
| (ENNReal.toReal_pos hvol.ne' (fillingVolume_lt_top L s).ne).ne' |
| have hdiff : IntegrableOn (fun a => filledHeight L s a y-filledHeight L s a x) |
| (fillingPolytope L s) := |
| (integrableOn_filledHeight L s y).sub (integrableOn_filledHeight L s x) |
| have hconst : IntegrableOn (fun _ : Filling L => edgeOffset L τ s x) (fillingPolytope L s) := |
| continuousOn_const.integrableOn_compact (isCompact_fillingPolytope L s) |
| rw [filling_volume_image E hE, hE.setIntegral_image_emb (linearCoordinates_measurableEmbedding E)] |
| simp_rw [hE0 s] |
| rw [integral_add (hdiff.const_mul τ) hconst, integral_const_mul, |
| integral_sub (integrableOn_filledHeight L s y) (integrableOn_filledHeight L s x),setIntegral_const] |
| simp only [meanHeight, smul_eq_mul, Measure.real] |
| change (_+ (fillingVolume L s).toReal*_) / (fillingVolume L s).toReal = _ |
| field_simp |
| |
| end TriangularLipschitz |
| |
| end |
| end |
| |
| end OAI |
| |
