| 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 | |