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0789dda | 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 | import Mathlib
namespace OAI
noncomputable section
open Set MeasureTheory
open scoped RealInnerProductSpace
namespace GeneralMahler
abbrev Rn (n : ℕ) := EuclideanSpace ℝ (Fin n)
/-- The polar of a convex body translated by an interior center. -/
def polarAt {n : ℕ} (K : Set (Rn n)) (z : Rn n) : Set (Rn n) :=
{y | ∀ x ∈ K, ⟪y, x - z⟫ ≤ 1}
/-- The volume product minimized over the interior centers of the body. -/
def volumeProduct {n : ℕ} (K : Set (EuclideanSpace ℝ (Fin n))) : ℝ :=
sInf ((fun z => (volume K).toReal * (volume (polarAt K z)).toReal) ''
interior K)
/-- The sharp general Mahler inequality, with equality exactly for simplices. -/
theorem general_mahler {n : ℕ} (hn : 1 ≤ n)
(K : Set (EuclideanSpace ℝ (Fin n)))
(hcompact : IsCompact K) (hconvex : Convex ℝ K)
(hinterior : (interior K).Nonempty) :
((n : ℝ) + 1) ^ (n + 1) / (Nat.factorial n : ℝ) ^ 2 ≤ volumeProduct K ∧
(volumeProduct K = ((n : ℝ) + 1) ^ (n + 1) / (Nat.factorial n : ℝ) ^ 2 ↔
∃ vertices : Fin (n + 1) → EuclideanSpace ℝ (Fin n),
AffineIndependent ℝ vertices ∧ K = convexHull ℝ (range vertices)) := by
sorry
end GeneralMahler
end
end OAI
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