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465b10d | 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 | import Mathlib
namespace OAI
noncomputable section
open Set MeasureTheory
open scoped RealInnerProductSpace
namespace PettyProjection
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
def unitBall (n : ℕ) : Set (Space n) := Metric.closedBall 0 1
def kappa (n : ℕ) : ℝ := volume.real (unitBall n)
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)
def pettyConstant (n : ℕ) : ℝ :=
kappa (n - 1) ^ n * kappa n ^ (2 - (n : ℤ))
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
/-- The exact unrestricted inequality and its global equality characterization.
-/
theorem petty_projection_volume (n : ℕ) (hn : 4 ≤ n)
(K : Set (Space n)) (hK : IsConvexBody K) :
pettyConstant n ≤ projectionRatio K ∧
(projectionRatio K = pettyConstant n ↔ IsEllipsoid K) := by
sorry
end PettyProjection
end
end OAI
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