File size: 8,761 Bytes
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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 | 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
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
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 ∈ interior L} ≃ Fin (n+1))
{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
|