| import OAI.Geometry.CylinderCovering.Barycentric.SweepsDefinitions |
| import OAI.Geometry.CylinderCovering.Radial.RiemannDefinitions |
|
|
| namespace OAI |
|
|
| open Set MeasureTheory Filter |
| open scoped Topology InnerProductSpace |
| noncomputable section |
|
|
| namespace RadialCoordinates.BarycentricFinite |
|
|
| abbrev Space := BarycentricSpace |
|
|
| def midpoint (N : ℕ) (i : Fin N) : ℝ := |
| (CylinderCovering.RadialRiemann.point 0 1 N i + |
| CylinderCovering.RadialRiemann.point 0 1 N (i + 1)) / 2 |
|
|
| def secondDirection (k t : ℝ) : Space := |
| !₂[(1 + k * angular t) / 2, (-1 + k * angular t) / 2, |
| -k * t^2, k * (1-t)^2] |
|
|
| def firstAxis (k t : ℝ) : Space := ‖firstDirection k t‖⁻¹ • firstDirection k t |
|
|
| def secondAxis (k t : ℝ) : Space := ‖secondDirection k t‖⁻¹ • secondDirection k t |
|
|
| def projection (u z : Space) : Space := z - inner ℝ u z • u |
|
|
| def firstLocalSweep (k : ℝ) (N : ℕ) (i : Fin N) : Set Space := |
| {z | ∃ t ∈ CylinderCovering.RadialRiemann.cell 0 1 N i, |
| ∃ S ∈ Icc (0 : ℝ) (cap k t), ∃ p ∈ Icc (-1 : ℝ) 1, |
| z = firstSweepPoint k t S p} |
|
|
| def secondLocalSweep (k : ℝ) (N : ℕ) (i : Fin N) : Set Space := |
| {z | ∃ t ∈ CylinderCovering.RadialRiemann.cell 0 1 N i, |
| ∃ S ∈ Icc (0 : ℝ) (cap k t), ∃ p ∈ Icc (-1 : ℝ) 1, |
| z = secondSweepPoint k t S p} |
|
|
| def firstBase (k : ℝ) (N : ℕ) (i : Fin N) : Set Space := |
| projection (firstAxis k (midpoint N i)) '' firstLocalSweep k N i |
|
|
| def secondBase (k : ℝ) (N : ℕ) (i : Fin N) : Set Space := |
| projection (secondAxis k (midpoint N i)) '' secondLocalSweep k N i |
|
|
| def cylinder (u : Space) (B : Set Space) : Set Space := |
| {z | ∃ b ∈ B, ∃ s : ℝ, z = b + s • u} |
|
|
| def area (B : Set Space) : ℝ := (Measure.euclideanHausdorffMeasure 2).real B |
|
|
| def totalArea (k : ℝ) (N : ℕ) : ℝ := |
| (∑ i : Fin N, area (firstBase k N i)) + ∑ i : Fin N, area (secondBase k N i) |
|
|
| def Statement : Prop := |
| ∀ k : ℝ, 0 < k → k ≤ 1 / 5000 → |
| (∀ N : ℕ, 0 < N → |
| (∀ i : Fin N, |
| ‖firstAxis k (midpoint N i)‖ = 1 ∧ |
| ‖secondAxis k (midpoint N i)‖ = 1 ∧ |
| IsCompact (firstBase k N i) ∧ IsCompact (secondBase k N i) ∧ |
| MeasurableSet (firstBase k N i) ∧ MeasurableSet (secondBase k N i) ∧ |
| Measure.euclideanHausdorffMeasure 2 (firstBase k N i) ≠ ⊤ ∧ |
| Measure.euclideanHausdorffMeasure 2 (secondBase k N i) ≠ ⊤ ∧ |
| (∀ z ∈ firstBase k N i, |
| inner ℝ (firstAxis k (midpoint N i)) z = 0 ∧ ∑ j, z j = 1) ∧ |
| (∀ z ∈ secondBase k N i, |
| inner ℝ (secondAxis k (midpoint N i)) z = 0 ∧ ∑ j, z j = 1)) ∧ |
| (∀ z ∈ barycentricSimplex, |
| (∃ i : Fin N, z ∈ cylinder (firstAxis k (midpoint N i)) (firstBase k N i)) ∨ |
| (∃ i : Fin N, z ∈ cylinder (secondAxis k (midpoint N i)) (secondBase k N i)))) ∧ |
| (∀ δ : ℝ, 0 < δ → ∀ᶠ N in atTop, totalArea k N ≤ 2 * J k + δ) ∧ |
| Filter.limsup (totalArea k) atTop ≤ 2 * J k ∧ |
| (∀ᶠ N in atTop, 0 < N ∧ totalArea k N < Real.sqrt 2 / 4) |
|
|
| end RadialCoordinates.BarycentricFinite |
|
|
| end |
|
|
| end OAI |
|
|
