File size: 4,232 Bytes
2fca478 | 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 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 | import OAI.Geometry.ProjectionBody.ProductBrightness
import OAI.Geometry.ProjectionBody.ProductZonotopeBridge
import OAI.Geometry.Zonotope.EuclideanVolume
import OAI.Geometry.ProjectionBody.Arithmetic
namespace OAI
/-!
The geometric projection formulas identify the two actual projection bodies.
Their proved Euclidean volumes are substituted into the normalized ratio before
the final exact arithmetic is performed.
-/
noncomputable section
open Set MeasureTheory
open scoped RealInnerProductSpace
namespace ProjectionCounterexample
theorem simplex_twenty_brightness {u : E 20} (hu : ‖u‖ = 1) :
projectionVolume (simplex 20) u =
(1 / (Nat.factorial 19 : ℝ)) * DiagonalZonotope.support (WithLp.ofLp u) := by
rw [simplex_projectionVolume (n := 19) (by decide) hu]
unfold DiagonalZonotope.support
ring
theorem product_brightness {u : E 20} (hu : ‖u‖ = 1) :
projectionVolume productBody u =
(1 / ((Nat.factorial 9 : ℝ) * Nat.factorial 10)) * blockSupport (n := 10) (m := 10) u := by
rw [product_projectionVolume hu]
change (((∑ i, |firstBlock u i|) + |∑ i, firstBlock u i|) +
((∑ i, |secondBlock u i|) + |∑ i, secondBlock u i|)) /
(2 * ((Nat.factorial 9 : ℝ) * Nat.factorial 10)) =
(1 / ((Nat.factorial 9 : ℝ) * Nat.factorial 10)) *
(((∑ i, |firstBlock u i|) + |∑ i, firstBlock u i|) / 2 +
((∑ i, |secondBlock u i|) + |∑ i, secondBlock u i|) / 2)
ring
theorem simplex_twenty_projectionBody :
projectionBody (simplex 20) =
DiagonalZonotope.euclidean (1 / (Nat.factorial 19 : ℝ)) :=
projectionBody_eq_zonotope (simplex 20) (1 / (Nat.factorial 19 : ℝ))
(by positivity) (fun _ => simplex_twenty_brightness)
theorem product_projectionBody :
projectionBody productBody = DiagonalZonotope.euclideanProduct 10 10
(1 / ((Nat.factorial 9 : ℝ) * Nat.factorial 10)) :=
projectionBody_eq_productZonotope (n := 10) (m := 10) productBody
(1 / ((Nat.factorial 9 : ℝ) * Nat.factorial 10))
(by positivity) (fun _ => product_brightness)
theorem simplex_twenty_projection_support :
IsConvexBody (projectionBody (simplex 20)) ∧
∀ u : E 20, ‖u‖ = 1 →
∃ x ∈ projectionBody (simplex 20), ⟪u, x⟫ = projectionVolume (simplex 20) u :=
projection_zonotope_is_body_and_attains (simplex 20) (1 / (Nat.factorial 19 : ℝ))
(by positivity)
(fun _ => simplex_twenty_brightness)
theorem product_projection_support :
IsConvexBody (projectionBody productBody) ∧
∀ u : E 20, ‖u‖ = 1 →
∃ x ∈ projectionBody productBody, ⟪u, x⟫ = projectionVolume productBody u :=
projection_productZonotope_is_body_and_attains (n := 10) (m := 10) productBody
(1 / ((Nat.factorial 9 : ℝ) * Nat.factorial 10)) (by positivity)
(fun _ => product_brightness)
theorem simplex_twenty_ratio :
ratio (simplex 20) = ((21 : ℝ) * 20 ^ 20) / (Nat.factorial 20 : ℝ) := by
unfold ratio bodyVolume
rw [simplex_twenty_projectionBody,
DiagonalZonotope.volume_euclidean_toReal _ (by positivity), volume_simplex]
norm_num [Nat.factorial]
theorem product_ratio_value :
ratio productBody = (((11 : ℝ) * 10 ^ 10) / (Nat.factorial 10 : ℝ)) ^ 2 := by
unfold ratio bodyVolume
rw [product_projectionBody,
DiagonalZonotope.volume_euclideanProduct_toReal 10 10 _ (by positivity), volume_productBody]
norm_num [Nat.factorial]
theorem product_to_simplex_ratio :
ratio productBody / ratio (simplex 20) = (22355476 : ℝ) / 22020096 := by
rw [product_ratio_value, simplex_twenty_ratio]
exact numerical_product_ratio
theorem product_exceeds_simplex : ratio (simplex 20) < ratio productBody := by
have hpos : 0 < ratio (simplex 20) := by
rw [simplex_twenty_ratio]
positivity
have hratio : 1 < ratio productBody / ratio (simplex 20) := by
rw [product_to_simplex_ratio]
exact numerical_strict_excess
simpa using (lt_div_iff₀ hpos).mp hratio
theorem universal_simplex_upper_bound_false :
¬ (∀ P : Set (E 20), IsConvexBody P → ratio P ≤ ratio (simplex 20)) := by
intro h
exact (not_le_of_gt product_exceeds_simplex) (h productBody productBody_convexBody)
end ProjectionCounterexample
end
end OAI
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