| import OAI.Geometry.ProjectionVolume.Arithmetic |
| import OAI.Geometry.ProjectionVolume.ProductBrightness |
| import OAI.Geometry.ProjectionVolume.ProductZonotope |
| import OAI.Geometry.ProjectionVolume.ProjectionReconstruction |
|
|
| namespace OAI |
|
|
| open MeasureTheory |
|
|
| namespace Paper092 |
|
|
| theorem productWitness_projectionBody : |
| projectionBody productWitness = productZonotope := by |
| apply projectionBody_eq_of_brightness_eq_support productZonotope_isCompact |
| productZonotope_nonempty productZonotope_convex |
| intro u |
| rw [productWitness_brightness, productZonotope_support] |
|
|
| theorem productZonotope_normalized_volume : |
| (volume productZonotope).toReal / (volume productWitness).toReal ^ 19 = |
| simplexConstant 10 * simplexConstant 10 := by |
| rw [productZonotope_volume_real, productWitness_volume_real] |
| norm_num [simplexConstant, Nat.factorial] |
|
|
| theorem productWitness_normalizedProjectionVolume : |
| normalizedProjectionVolume productWitness = simplexConstant 10 * simplexConstant 10 := by |
| unfold normalizedProjectionVolume |
| rw [productWitness_projectionBody] |
| exact productZonotope_normalized_volume |
|
|
| theorem product_counterexample : |
| (IsCompact productWitness ∧ Convex ℝ productWitness ∧ |
| (interior productWitness).Nonempty) ∧ |
| normalizedProjectionVolume productWitness / simplexConstant 20 = |
| (121 * (Nat.choose 20 10 : ℝ)) / (21 * (2 : ℝ) ^ 20) ∧ |
| normalizedProjectionVolume productWitness / simplexConstant 20 = |
| (22355476 : ℝ) / 22020096 ∧ |
| 1 < normalizedProjectionVolume productWitness / simplexConstant 20 ∧ |
| (volume (projectionBody productWitness)).toReal > |
| simplexConstant 20 * (volume productWitness).toReal ^ 19 := by |
| have hratio : normalizedProjectionVolume productWitness / simplexConstant 20 = |
| (22355476 : ℝ) / 22020096 := by |
| rw [productWitness_normalizedProjectionVolume] |
| exact dimension_twenty_ratio |
| have hchoose : normalizedProjectionVolume productWitness / simplexConstant 20 = |
| (121 * (Nat.choose 20 10 : ℝ)) / (21 * (2 : ℝ) ^ 20) := by |
| rw [hratio, Nat.choose_eq_factorial_div_factorial (by norm_num : 10 ≤ 20)] |
| norm_num [Nat.factorial] |
| have hgt : 1 < normalizedProjectionVolume productWitness / simplexConstant 20 := by |
| rw [hratio] |
| norm_num |
| have hnorm : simplexConstant 20 < normalizedProjectionVolume productWitness := by |
| rw [productWitness_normalizedProjectionVolume] |
| exact dimension_twenty_strict |
| change simplexConstant 20 < |
| (volume (projectionBody productWitness)).toReal / (volume productWitness).toReal ^ 19 at hnorm |
| have hvol := (lt_div_iff₀ (pow_pos productWitness_volume_pos 19)).mp hnorm |
| exact ⟨productWitness_is_convex_body, hchoose, hratio, hgt, hvol⟩ |
|
|
| end Paper092 |
|
|
| end OAI |
|
|
