| import Mathlib | |
| namespace OAI | |
| noncomputable section | |
| open Set MeasureTheory | |
| open scoped ENNReal InnerProductSpace | |
| namespace LogBrunnMinkowski | |
| 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 OriginSymmetric {n : ℕ} (K : Set (Space n)) : Prop := | |
| ∀ x, x ∈ K ↔ -x ∈ K | |
| def support {n : ℕ} (K : Set (Space n)) (u : Space n) : ℝ := | |
| sSup ((fun x : Space n => ⟪x, u⟫_ℝ) '' K) | |
| def wulff {n : ℕ} (f : Space n → ℝ) : Set (Space n) := | |
| {x | ∀ u, ‖u‖ = 1 → ⟪x, u⟫_ℝ ≤ f u} | |
| def logCombination {n : ℕ} (K L : Set (Space n)) (t : ℝ) : | |
| Set (Space n) := | |
| wulff (fun u => (support K u) ^ (1 - t) * (support L u) ^ t) | |
| theorem main (n : ℕ) (hn : 1 ≤ n) (K L : Set (Space n)) | |
| (hK : IsConvexBody K) (hL : IsConvexBody L) | |
| (hsK : OriginSymmetric K) (hsL : OriginSymmetric L) | |
| (t : ℝ) (ht0 : 0 ≤ t) (ht1 : t ≤ 1) : | |
| (volume K) ^ (1 - t) * (volume L) ^ t ≤ volume (logCombination K L t) := by | |
| sorry | |
| end LogBrunnMinkowski | |
| end | |
| end OAI | |