| import Mathlib |
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|
| namespace OAI |
|
|
| noncomputable section |
| open Set MeasureTheory |
| open scoped RealInnerProductSpace |
|
|
| namespace PettyProjection |
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|
| abbrev Space (n : ℕ) := EuclideanSpace ℝ (Fin n) |
|
|
| def IsConvexBody {n : ℕ} (K : Set (Space n)) : Prop := |
| IsCompact K ∧ Convex ℝ K ∧ (interior K).Nonempty |
|
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| def unitBall (n : ℕ) : Set (Space n) := Metric.closedBall 0 1 |
|
|
| def kappa (n : ℕ) : ℝ := volume.real (unitBall n) |
|
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| def perpendicular {n : ℕ} (u : Space n) : Submodule ℝ (Space n) := |
| (Submodule.span ℝ {u})ᗮ |
|
|
| /-- Volume in the induced Euclidean structure on the orthogonal hyperplane. -/ |
| def shadowVolume {n : ℕ} (K : Set (Space n)) (u : Space n) : ℝ := |
| (volume : Measure (perpendicular u)).real |
| ((perpendicular u).orthogonalProjectionOnto '' K) |
|
|
| /-- The body with support function equal to the volumes of orthogonal shadows, |
| expressed as its defining intersection of supporting half-spaces. -/ |
| def projectionBody {n : ℕ} (K : Set (Space n)) : Set (Space n) := |
| {x | ∀ u : Space n, ‖u‖ = 1 → ⟪u, x⟫ ≤ shadowVolume K u} |
|
|
| def projectionRatio {n : ℕ} (K : Set (Space n)) : ℝ := |
| volume.real (projectionBody K) / volume.real K ^ (n - 1) |
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|
| def pettyConstant (n : ℕ) : ℝ := |
| kappa (n - 1) ^ n * kappa n ^ (2 - (n : ℤ)) |
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| def IsEllipsoid {n : ℕ} (K : Set (Space n)) : Prop := |
| ∃ (a : Space n) (T : Space n ≃ₗ[ℝ] Space n), |
| K = (fun x => a + T x) '' unitBall n |
|
|
| def MainClaim : Prop := |
| ∀ (n : ℕ), 4 ≤ n → ∀ (K : Set (Space n)), IsConvexBody K → |
| pettyConstant n ≤ projectionRatio K ∧ |
| (projectionRatio K = pettyConstant n ↔ IsEllipsoid K) |
|
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| end PettyProjection |
| end |
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| end OAI |
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|
