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dd14c26 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 | 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
noncomputable section
namespace TriangularLipschitz
end TriangularLipschitz
namespace FacetExposure
end FacetExposure
namespace TriangularLipschitz
end TriangularLipschitz
namespace TriangularLipschitz
end TriangularLipschitz
namespace TriangularLipschitz
end TriangularLipschitz
open MeasureTheory Set
open scoped ENNReal
noncomputable section
namespace EdgeCoordinates
end EdgeCoordinates
namespace EdgeCoordinates
end EdgeCoordinates
namespace EdgeCoordinates
end EdgeCoordinates
namespace TriangularLipschitz
end TriangularLipschitz
namespace TriangularLipschitz
end TriangularLipschitz
namespace TriangularLipschitz
end TriangularLipschitz
open Set MeasureTheory Filter
open scoped Topology
noncomputable section
namespace FacetVolume
end FacetVolume
open Set MeasureTheory Filter
open scoped Topology
noncomputable section
namespace SliceTransport
end SliceTransport
namespace FacetVolume
end FacetVolume
namespace FacetVolume
end FacetVolume
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
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