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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
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
end TriangularLipschitz
open scoped BigOperators
namespace TriangularLipschitz
end TriangularLipschitz
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
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