| import OAI.Geometry.ProjectionBody.Counterexample | |
| namespace OAI | |
| open Set | |
| open scoped RealInnerProductSpace | |
| namespace ProjectionCounterexample.Specification | |
| theorem product_is_convex_body : IsConvexBody productBody := productBody_convexBody | |
| theorem projection_support_attained : | |
| ∀ P : Set (E 20), P = simplex 20 ∨ P = productBody → | |
| IsConvexBody (projectionBody P) ∧ | |
| ∀ u : E 20, ‖u‖ = 1 → | |
| ∃ x ∈ projectionBody P, ⟪u, x⟫ = projectionVolume P u := by | |
| intro P hP | |
| rcases hP with rfl | rfl | |
| · exact simplex_twenty_projection_support | |
| · exact product_projection_support | |
| theorem simplex_ratio : | |
| ratio (simplex 20) = ((21 : ℝ) * (20 : ℝ) ^ 20) / (Nat.factorial 20 : ℝ) := | |
| simplex_twenty_ratio | |
| theorem product_ratio : | |
| ratio productBody / ratio (simplex 20) = (22355476 : ℝ) / 22020096 ∧ | |
| (1 : ℝ) < ratio productBody / ratio (simplex 20) := by | |
| constructor | |
| · exact product_to_simplex_ratio | |
| · rw [product_to_simplex_ratio] | |
| exact numerical_strict_excess | |
| theorem simplex_upper_bound_false : | |
| ¬ (∀ P : Set (E 20), IsConvexBody P → ratio P ≤ ratio (simplex 20)) := | |
| universal_simplex_upper_bound_false | |
| end ProjectionCounterexample.Specification | |
| end OAI | |