File size: 2,749 Bytes
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 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 | 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
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