File size: 2,140 Bytes
cfa2fd2 | 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 | import Mathlib
namespace OAI
open Set MeasureTheory
open scoped InnerProductSpace BigOperators Interval
noncomputable section
namespace TriangularCovering
abbrev Space := EuclideanSpace ℝ (Fin 3)
def H : ℝ := Real.sqrt 2
def point (x y t : ℝ) : Space := !₂[x, y, H * t]
def K : Set Space :=
{z | ∃ x y t : ℝ, 0 ≤ t ∧ t ≤ 1 ∧ |x| ≤ 1 - t ∧ |y| ≤ t ∧ z = point x y t}
def projection (u z : Space) : Space := z - (inner ℝ u z) • u
def area (B : Set Space) : ℝ := (Measure.euclideanHausdorffMeasure 2 B).toReal
def shadowArea (u : Space) : ℝ := area (projection u '' K)
def A_min : ℝ := sInf {a : ℝ | ∃ u : Space, ‖u‖ = 1 ∧ a = shadowArea u}
structure TriCylinder where
axis : Space
base : Set Space
unit_axis : ‖axis‖ = 1
perpendicular : ∀ z ∈ base, inner ℝ axis z = 0
triangular : ∃ v : Fin 3 → Space, AffineIndependent ℝ v ∧
base = convexHull ℝ (Set.range v)
compact : IsCompact base
convex : Convex ℝ base
measurable : MeasurableSet base
finite_area : Measure.euclideanHausdorffMeasure 2 base ≠ ⊤
def TriCylinder.carrier (c : TriCylinder) : Set Space :=
{z | ∃ b ∈ c.base, ∃ r : ℝ, z = b + r • c.axis}
def Covers {m : ℕ} (c : Fin m → TriCylinder) : Prop :=
∀ z ∈ K, ∃ i : Fin m, z ∈ (c i).carrier
def totalArea {m : ℕ} (c : Fin m → TriCylinder) : ℝ :=
∑ i : Fin m, area (c i).base
def relativeCost {m : ℕ} (c : Fin m → TriCylinder) : ℝ :=
∑ i : Fin m, area (c i).base / shadowArea (c i).axis
def n (ε : ℝ) : ℕ := ⌈2 / ε ^ 2⌉₊
def MainAsymptotic : Prop :=
∃ δ C : ℝ, 0 < δ ∧ 0 < C ∧
∀ ε : ℝ, 0 < ε → ε < δ →
∃ c : Fin (2 * n ε) → TriCylinder,
Covers c ∧
|totalArea c / H - (1 / 2 - (13 / 6000 : ℝ) * ε ^ 2)| ≤ C * ε ^ 4 ∧
totalArea c < A_min / 2
def NormalizedCounterexample : Prop :=
∃ m : ℕ, ∃ c : Fin m → TriCylinder, Covers c ∧ relativeCost c < 1 / 2
theorem main : A_min = H ∧ MainAsymptotic ∧ NormalizedCounterexample := by
sorry
end TriangularCovering
end
end OAI
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