| import Mathlib |
| import OAI.Probability.LatticeSurface.Convexity.BrunnMinkowski |
| import OAI.Probability.LatticeSurface.Convexity.CubePrekopa |
| import OAI.Probability.LatticeSurface.Convexity.FacetDeficitAnalytic |
| import OAI.Probability.LatticeSurface.Convexity.FacetQuantile |
| import OAI.Probability.LatticeSurface.Convexity.InteriorQuantile |
| import OAI.Probability.LatticeSurface.Convexity.QuantileCentroid |
| import OAI.Probability.LatticeSurface.Convexity.QuantileTransport |
| import OAI.Probability.LatticeSurface.Convexity.QuantitativeTransport |
| import OAI.Probability.LatticeSurface.Convexity.RadialExposure |
| import OAI.Probability.LatticeSurface.Convexity.SliceDensity |
| import OAI.Probability.LatticeSurface.Convexity.SliceGeometry |
| import OAI.Probability.LatticeSurface.Convexity.SliceMoments |
| import OAI.Probability.LatticeSurface.Convexity.WeightedPrekopa |
| import OAI.Probability.LatticeSurface.Dynamics.DisplacementInequality |
| import OAI.Probability.LatticeSurface.Geometry.BlockAnalytic |
| import OAI.Probability.LatticeSurface.Geometry.BlockEdgeGeometry |
| import OAI.Probability.LatticeSurface.Geometry.BlockFacetGeometry |
| import OAI.Probability.LatticeSurface.Geometry.BlockRadial |
| import OAI.Probability.LatticeSurface.Geometry.BlockTelescope |
| import OAI.Probability.LatticeSurface.Geometry.BlockVolume |
| import OAI.Probability.LatticeSurface.Geometry.EdgeCoordinates |
| import OAI.Probability.LatticeSurface.Geometry.FacetBody |
| import OAI.Probability.LatticeSurface.Geometry.FacetExposure |
| import OAI.Probability.LatticeSurface.Geometry.TriangularCover |
| import OAI.Probability.LatticeSurface.Geometry.TriangularGeometry |
| import OAI.Probability.LatticeSurface.Geometry.UniformFacet |
|
|
| namespace OAI |
|
|
| open MeasureTheory |
|
|
| 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 |
|
|
| 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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| 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 |
|
|
| open Set MeasureTheory Filter |
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| open scoped Topology |
|
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| noncomputable section |
|
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| namespace FacetVolume |
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| end FacetVolume |
|
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| open Set MeasureTheory Filter |
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| open scoped Topology |
|
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| noncomputable section |
|
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| namespace SliceTransport |
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| end SliceTransport |
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| namespace FacetVolume |
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| end FacetVolume |
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| namespace FacetVolume |
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| end FacetVolume |
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| namespace TriangularLipschitz |
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| end TriangularLipschitz |
|
|
| open scoped BigOperators |
|
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| namespace TriangularLipschitz |
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| end TriangularLipschitz |
|
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| namespace TriangularLipschitz |
|
|
| lemma fillingVolume_shrunk_pos {L : ℕ} {h χ : Vertex → ℝ} {δ α : ℝ} |
| (hδ : 0 < δ) (hα : 0 ≤ α) (hα1 : α < 1) |
| (hne : ∀ x ∈ box L, ∀ y ∈ box L, Adj x y → |h x-h y| ≤ 1-δ) |
| (hplus : 0 < fillingVolume L (fun x => h x+χ x)) : |
| 0 < fillingVolume L (fun x => h x+α*χ x) := by |
| have heq : (fun x => (1-α)*h x+α*(h x+χ x))=(fun x => h x+α*χ x) := by funext x; ring |
| rw [← heq] |
| exact fillingVolume_interpolated_pos hδ hα hα1 hne (nonempty_of_measure_ne_zero hplus.ne') |
| |
| lemma shrunk_deficit_le {L n : ℕ} (E : Filling L ≃ₗ[ℝ] (Fin n → ℝ)) (hE : MeasurePreserving E) |
| {h χ : Vertex → ℝ} {α : ℝ} (hα : 0 ≤ α) (hα1 : α ≤ 1) |
| (hh : 0 < fillingVolume L h) |
| (hplus : 0 < fillingVolume L (fun x => h x+χ x)) |
| (hminus : 0 < fillingVolume L (fun x => h x-χ x)) : |
| positiveLogVolume L h-(positiveLogVolume L (fun x => h x+α*χ x)+ |
| positiveLogVolume L (fun x => h x-α*χ x))/2 ≤ |
| positiveLogVolume L h-(positiveLogVolume L (fun x => h x+χ x)+ |
| positiveLogVolume L (fun x => h x-χ x))/2 := by |
| have hp := positiveLogVolume_concave E hE h (fun x => h x+χ x) hh hplus |
| (by linarith : 0 ≤ 1-α) hα (by ring : (1-α)+α=1) |
| have hm := positiveLogVolume_concave E hE h (fun x => h x-χ x) hh hminus |
| (by linarith : 0 ≤ 1-α) hα (by ring : (1-α)+α=1) |
| have hmid := positiveLogVolume_concave E hE (fun x => h x+χ x) (fun x => h x-χ x) |
| hplus hminus (by norm_num : (0:ℝ) ≤ 1/2) (by norm_num : (0:ℝ) ≤ 1/2) |
| (by norm_num : (1/2:ℝ)+1/2=1) |
| have hfp : (fun x => (1-α)*h x+α*(h x+χ x))=(fun x => h x+α*χ x) := by funext x; ring |
| have hfm : (fun x => (1-α)*h x+α*(h x-χ x))=(fun x => h x-α*χ x) := by funext x; ring |
| have hfmid : (fun x => (1/2:ℝ)*(h x+χ x)+(1/2:ℝ)*(h x-χ x))=h := by funext x; ring |
| rw [hfp] at hp |
| rw [hfm] at hm |
| rw [hfmid] at hmid |
| have hg : 0 ≤ positiveLogVolume L h-(positiveLogVolume L (fun x => h x+χ x)+ |
| positiveLogVolume L (fun x => h x-χ x))/2 := by linarith |
| have hmul := mul_le_mul_of_nonneg_right hα1 hg |
| nlinarith |
| |
| theorem triangular_block_deficit |
| (δ δ₀ s₀ : ℝ) (k₀ L : ℕ) |
| (hδ : 0 < δ) (hδδ₀ : δ ≤ δ₀) (hδ₀ : δ₀ ≤ 1 / 16) |
| (hk₀ : 4 / δ₀ ≤ (k₀ : ℝ)) (hs₀ : 0 < s₀) |
| (hLeven : Even L) (hL : 8 * ((k₀ : ℝ) + 2) / δ₀ ≤ (L : ℝ)) : |
| ∃ c : ℝ, 0 < c ∧ ∀ h χ : Vertex → ℝ, |
| ConvexGood L k₀ δ δ₀ s₀ h → |
| 0 < fillingVolume L (fun x => h x + χ x) → |
| 0 < fillingVolume L (fun x => h x - χ x) → |
| ∀ v w : Fin 4, |
| c * |χ (faceCenter L v) - χ (faceCenter L w)| ^ 2 ≤ |
| positiveLogVolume L h - |
| (positiveLogVolume L (fun x => h x + χ x) + |
| positiveLogVolume L (fun x => h x - χ x)) / 2 := by |
| classical |
| obtain ⟨β,hβ,hβ1,hβδ,hβs,hexpose⟩ := |
| triangular_radial_exposure δ δ₀ s₀ k₀ L hδ hδδ₀ hδ₀ hk₀ hs₀ hLeven hL |
| have hδ₀pos := hδ.trans_le hδδ₀ |
| have hLposR : (0:ℝ) < L := lt_of_lt_of_le (div_pos (by positivity) hδ₀pos) hL |
| have hLpos : 0 < L := by exact_mod_cast hLposR |
| obtain ⟨m,hm⟩ := hLeven |
| have hLeq : L=2*m := by omega |
| clear hm |
| subst L |
| have hm : 0 < m := by omega |
| let α : ℝ := β/(16*((2*m:ℕ)+1)) |
| have hα : 0 < α := div_pos hβ (by positivity) |
| have hα1 : α < 1 := by |
| apply (div_lt_iff₀ (by positivity : (0:ℝ) < 16*((2*m:ℕ)+1))).mpr |
| have hnonneg : (0:ℝ) ≤ (2*m:ℕ) := Nat.cast_nonneg _ |
| linarith |
| have hαβ : |α| * (2*((2*m:ℕ):ℝ)) ≤ β/4 := by |
| rw [abs_of_pos hα] |
| dsimp [α] |
| rw [div_mul_eq_mul_div] |
| apply (div_le_iff₀ (by positivity : (0:ℝ) < 16*((2*m:ℕ)+1))).mpr |
| nlinarith [hβ] |
| have hαβneg : |(-α)| * (2*((2*m:ℕ):ℝ)) ≤ β/4 := by simpa only [abs_neg] using hαβ |
| obtain ⟨n,e,he⟩ := EdgeCoordinates.exists_indexEquiv |
| (⟨radialPoint (2*m) 0 1,radialPoint_mem_interior (by omega) (by omega : 1 ≤ m) 0⟩ : |
| {x : Vertex // x ∈ interior (2*m)}) |
| let E₀ := EdgeCoordinates.reindex e |
| have hE₀ : MeasurePreserving E₀ := EdgeCoordinates.reindex_measurePreserving e |
| obtain ⟨κ,hκ,hfacet⟩ := exists_filling_facet_constant (2*m) n β hβ |
| have hmR : (0:ℝ) < m := by exact_mod_cast hm |
| have hmsq : (0:ℝ) < (m:ℝ)^2 := sq_pos_of_pos hmR |
| refine ⟨κ*α^2/(m:ℝ)^2,by positivity,?_⟩ |
| intro h χ hgood hplus hminus v w |
| have hh : 0 < fillingVolume (2*m) h := fillingVolume_pos_of_strict hδ hgood.1 |
| let s : Vertex → ℝ := fun x => h x+α*χ x |
| let t : Vertex → ℝ := fun x => h x-α*χ x |
| have hs : 0 < fillingVolume (2*m) s := fillingVolume_shrunk_pos hδ hα.le hα1 hgood.1 hplus |
| have ht : 0 < fillingVolume (2*m) t := by |
| have hm' : 0 < fillingVolume (2*m) (fun x => h x+(-χ x)) := by simpa only [sub_eq_add_neg] using hminus |
| simpa only [t,mul_neg,sub_eq_add_neg] using fillingVolume_shrunk_pos hδ hα.le hα1 hgood.1 hm' |
| let U : Vertex → ℝ := fun x => meanHeight (2*m) s x-meanHeight (2*m) t x |
| let gα := positiveLogVolume (2*m) h-(positiveLogVolume (2*m) s+positiveLogVolume (2*m) t)/2 |
| have hedge : ∀ side : Fin 4, ∀ k < m, |
| κ*(U (radialPoint (2*m) side (k+1))-U (radialPoint (2*m) side k))^2 ≤ gα := by |
| intro side k hk |
| obtain ⟨g,hg,hgrad,hother⟩ := hexpose h hgood side k (by simpa using hk) |
| let x := radialPoint (2*m) side k |
| let y := radialPoint (2*m) side (k+1) |
| let τ := g y-g x |
| have hτ : |τ|=1 := by simpa only [τ,x,y,abs_sub_comm] using hgrad |
| have hxy : Adj x y := radialPoint_adj (2*m) k side |
| obtain ⟨E,hE,hEn,hE0⟩ := exists_filling_edge_coordinates e |
| (radialPoint_mem_interior (by omega : 0 < k+1) (by omega : k+1 ≤ m) side) (adj_ne hxy) hτ |
| have hball (σ : ℝ) (hσ : |σ| * (2*((2*m:ℕ):ℝ)) ≤ β/4) : |
| ∃ q₀ : Fin n → ℝ, ∀ q ∈ Metric.ball q₀ (β/16), |
| Fin.cons (1+edgeOffset (2*m) τ (fun b => h b+σ*χ b) x) q ∈ |
| E '' fillingPolytope (2*m) (fun b => h b+σ*χ b) := by |
| apply filling_facet_ball hτ hβ E hEn hE0 (fun z => g z+σ*χ (faceCenter (2*m) side)) |
| · exact shifted_radial_facet hm hk side hg hgrad |
| · exact shifted_exposure hβ |
| (shifted_shell_error hβ (faceCenter_mem_shell hm side) hg hplus hminus hσ) hother |
| have hbp := hball α hαβ |
| have hbm := hball (-α) hαβneg |
| have hbm' : ∃ q₀ : Fin n → ℝ, ∀ q ∈ Metric.ball q₀ (β/16), |
| Fin.cons (1+edgeOffset (2*m) τ t x) q ∈ E '' fillingPolytope (2*m) t := by |
| simpa only [neg_mul,← sub_eq_add_neg] using hbm |
| have hresult := hfacet s t h x y τ E hE hτ |
| (radialPoint_mem_box (Nat.le_of_lt hk) side) (radialPoint_mem_box (by omega : k+1 ≤ m) side) |
| hxy hE0 hs ht hh hbp hbm' (by intro b; dsimp [s,t]; ring) |
| convert hresult using 1; dsimp [U,gα,x,y]; ring |
| have hends (side : Fin 4) : U (radialPoint (2*m) side 0)=2*α*χ (faceCenter (2*m) side) := by |
| rw [radialPoint_zero] |
| have hfix := (shell_eq_box_not_interior.mp (faceCenter_mem_shell hm side)).2 |
| dsimp [U] |
| rw [meanHeight_fixed hs hfix,meanHeight_fixed ht hfix] |
| dsimp [s,t] |
| ring |
| have hbound := two_radial_square_bound (fun k => U (radialPoint (2*m) v k)) |
| (fun k => U (radialPoint (2*m) w k)) m hκ.le (by rw [radialPoint_center,radialPoint_center]) |
| (hedge v) (hedge w) |
| rw [hends v,hends w] at hbound |
| have hga := shrunk_deficit_le E₀ hE₀ hα.le hα1.le hh hplus hminus |
| calc |
| κ*α^2/(m:ℝ)^2 * |χ (faceCenter (2*m) v)-χ (faceCenter (2*m) w)|^2 = |
| (κ*α^2*|χ (faceCenter (2*m) v)-χ (faceCenter (2*m) w)|^2)/(m:ℝ)^2 := by ring |
| _ ≤ gα := (div_le_iff₀ hmsq).mpr (by |
| rw [sq_abs] |
| nlinarith only [hbound]) |
| _ ≤ _ := hga |
| |
| end TriangularLipschitz |
| |
| end |
| end |
| end |
| end |
| |
| end OAI |
| |
