| 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 |
|
|
