| 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.BlockEdgeGeometry |
| import OAI.Probability.LatticeSurface.Geometry.BlockFacetGeometry |
| import OAI.Probability.LatticeSurface.Geometry.BlockRadial |
| 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 |
|
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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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| 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 |
|
|
| lemma filling_image_mix {L n : ℕ} (E : Filling L ≃ₗ[ℝ] (Fin n → ℝ)) |
| (s t : Vertex → ℝ) {α γ : ℝ} (hα : 0 ≤ α) (hγ : 0 ≤ γ) (hsum : α+γ=1) : |
| ∀ x ∈ E '' fillingPolytope L s, ∀ y ∈ E '' fillingPolytope L t, |
| α • x+γ • y ∈ E '' fillingPolytope L (fun b => α*s b+γ*t b) := by |
| rintro x ⟨a,ha,rfl⟩ y ⟨b,hb,rfl⟩ |
| exact ⟨α • a+γ • b,fillingPolytope_mix ha hb hα hγ hsum,by simp⟩ |
|
|
| lemma positiveLogVolume_concave {L n : ℕ} (E : Filling L ≃ₗ[ℝ] (Fin n → ℝ)) |
| (hE : MeasurePreserving E) (s t : Vertex → ℝ) {α γ : ℝ} |
| (hs : 0 < fillingVolume L s) (ht : 0 < fillingVolume L t) |
| (hα : 0 ≤ α) (hγ : 0 ≤ γ) (hsum : α+γ=1) : |
| α*positiveLogVolume L s+γ*positiveLogVolume L t ≤ |
| positiveLogVolume L (fun b => α*s b+γ*t b) := by |
| have hs' : 0 < (fillingVolume L s).toReal := ENNReal.toReal_pos hs.ne' (fillingVolume_lt_top L s).ne |
| have ht' : 0 < (fillingVolume L t).toReal := ENNReal.toReal_pos ht.ne' (fillingVolume_lt_top L t).ne |
| have hbm := FacetVolume.compact_volume_brunn_minkowski |
| (isCompact_filling_image E s) (isCompact_filling_image E t) |
| (isCompact_filling_image E (fun b => α*s b+γ*t b)) |
| ((nonempty_of_measure_ne_zero hs.ne').image E) ((nonempty_of_measure_ne_zero ht.ne').image E) |
| hα hγ hsum (filling_image_mix E s t hα hγ hsum) |
| simp only [filling_volume_image E hE] at hbm |
| have hlog := Real.log_le_log (mul_pos (Real.rpow_pos_of_pos hs' α) (Real.rpow_pos_of_pos ht' γ)) hbm |
| rw [Real.log_mul (Real.rpow_pos_of_pos hs' α).ne' (Real.rpow_pos_of_pos ht' γ).ne', |
| Real.log_rpow hs',Real.log_rpow ht'] at hlog |
| exact hlog |
|
|
| lemma log_volume_ratio {L : ℕ} {s t h : Vertex → ℝ} |
| (hs : 0 < fillingVolume L s) (ht : 0 < fillingVolume L t) (hh : 0 < fillingVolume L h) : |
| Real.log ((fillingVolume L h).toReal/ |
| Real.sqrt ((fillingVolume L s).toReal*(fillingVolume L t).toReal)) = |
| positiveLogVolume L h-(positiveLogVolume L s+positiveLogVolume L t)/2 := by |
| have hs' : 0 < (fillingVolume L s).toReal := ENNReal.toReal_pos hs.ne' (fillingVolume_lt_top L s).ne |
| have ht' : 0 < (fillingVolume L t).toReal := ENNReal.toReal_pos ht.ne' (fillingVolume_lt_top L t).ne |
| have hh' : 0 < (fillingVolume L h).toReal := ENNReal.toReal_pos hh.ne' (fillingVolume_lt_top L h).ne |
| rw [Real.log_div hh'.ne' (Real.sqrt_pos.2 (mul_pos hs' ht')).ne', |
| Real.log_sqrt (mul_pos hs' ht').le,Real.log_mul hs'.ne' ht'.ne'] |
| rfl |
| |
| lemma exists_filling_facet_constant (L n : ℕ) (β : ℝ) (hβ : 0 < β) : |
| ∃ κ : ℝ, 0 < κ ∧ ∀ (s t h : Vertex → ℝ) (x y : Vertex) (τ : ℝ) |
| (E : Filling L ≃ₗ[ℝ] (Fin (n+1) → ℝ)), |
| MeasurePreserving E → |τ|=1 → x ∈ box L → y ∈ box L → Adj x y → |
| (∀ s a, E a 0 = τ*(filledHeight L s a y-filledHeight L s a x)+edgeOffset L τ s x) → |
| 0 < fillingVolume L s → 0 < fillingVolume L t → 0 < fillingVolume L h → |
| (∃ q₀ : Fin n → ℝ, ∀ q ∈ Metric.ball q₀ (β/16), |
| Fin.cons (1+edgeOffset L τ s x) q ∈ E '' fillingPolytope L s) → |
| (∃ q₀ : Fin n → ℝ, ∀ q ∈ Metric.ball q₀ (β/16), |
| Fin.cons (1+edgeOffset L τ t x) q ∈ E '' fillingPolytope L t) → |
| (∀ b, (s b+t b)/2=h b) → |
| κ*((meanHeight L s y-meanHeight L s x)-(meanHeight L t y-meanHeight L t x))^2 ≤ |
| positiveLogVolume L h-(positiveLogVolume L s+positiveLogVolume L t)/2 := by |
| obtain ⟨κ,hκ,hfacet⟩ := FacetVolume.exists_uniform_facet_constant n (β/16) |
| (volume (Metric.closedBall (0 : Filling L) (2*L))).toReal (by linarith) |
| refine ⟨κ,hκ,?_⟩ |
| intro s t h x y τ E hE hτ hx hy hxy hE0 hs ht hh hballS hballT hmid |
| obtain ⟨qS,hqS⟩ := hballS |
| obtain ⟨qT,hqT⟩ := hballT |
| have hresult := hfacet (E '' fillingPolytope L s) (E '' fillingPolytope L t) |
| (E '' fillingPolytope L h) (1+edgeOffset L τ s x) (1+edgeOffset L τ t x) qS qT |
| (isCompact_filling_image E s) (isCompact_filling_image E t) (isCompact_filling_image E h) |
| (convex_filling_image E s) (convex_filling_image E t) (convex_filling_image E h) |
| (by rwa [filling_volume_image E hE]) (by rwa [filling_volume_image E hE]) |
| (filling_projection_width s hτ hx hy hxy E hE0) (filling_projection_width t hτ hx hy hxy E hE0) |
| hqS hqT (filling_image_volume_bound E hE s) (filling_image_volume_bound E hE t) ?_ |
| · rw [coordinate_meanHeight E hE hE0 s hs,coordinate_meanHeight E hE hE0 t ht] at hresult |
| simp only [filling_volume_image E hE] at hresult |
| rw [log_volume_ratio hs ht hh] at hresult |
| have hτsq : τ^2=1 := by nlinarith [sq_abs τ] |
| have heq : ((1+edgeOffset L τ s x-(τ*(meanHeight L s y-meanHeight L s x)+edgeOffset L τ s x))- |
| (1+edgeOffset L τ t x-(τ*(meanHeight L t y-meanHeight L t x)+edgeOffset L τ t x)))^2 = |
| ((meanHeight L s y-meanHeight L s x)-(meanHeight L t y-meanHeight L t x))^2 := by |
| calc |
| _ = τ^2*((meanHeight L s y-meanHeight L s x)-(meanHeight L t y-meanHeight L t x))^2 := by ring |
| _ = _ := by rw [hτsq,one_mul] |
| rwa [heq] at hresult |
| · have heq : (fun b => (1/2:ℝ)*s b+(1/2:ℝ)*t b)=h := by |
| funext b; linarith [hmid b] |
| simpa only [heq] using filling_image_mix E s t (by norm_num : (0:ℝ) ≤ 1/2) |
| (by norm_num : (0:ℝ) ≤ 1/2) (by norm_num : (1/2:ℝ)+1/2=1) |
| |
| end TriangularLipschitz |
| |
| end |
| end |
| end |
| end |
| |
| end OAI |
| |
