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cead56f | 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 | 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
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