| import Mathlib |
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| namespace OAI |
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| noncomputable section |
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| open Set MeasureTheory |
| open scoped RealInnerProductSpace Pointwise |
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| namespace Paper092 |
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| noncomputable def simplexConstant (d : ℕ) : ℝ := |
| (d + 1) * (d : ℝ) ^ d / d.factorial |
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| abbrev Euclidean (n : ℕ) := EuclideanSpace ℝ (Fin n) |
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| def normalHyperplane {n : ℕ} (u : Euclidean n) : Submodule ℝ (Euclidean n) := |
| (Submodule.span ℝ ({u} : Set (Euclidean n)))ᗮ |
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| def projectionVolume {n : ℕ} (K : Set (Euclidean n)) (u : Euclidean n) : ENNReal := |
| (volume : Measure (normalHyperplane u)) |
| ((normalHyperplane u).orthogonalProjectionOnto '' K) |
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| def brightness {n : ℕ} (K : Set (Euclidean n)) (u : Euclidean n) : ℝ := |
| ‖u‖ * (projectionVolume K u).toReal |
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| def projectionBody {n : ℕ} (K : Set (Euclidean n)) : Set (Euclidean n) := |
| {y | ∀ u, ⟪u, y⟫ ≤ brightness K u} |
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| def normalizedProjectionVolume {n : ℕ} (K : Set (Euclidean n)) : ℝ := |
| (volume (projectionBody K)).toReal / (volume K).toReal ^ (n - 1) |
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| def standardSimplex (n : ℕ) : Set (Euclidean n) := |
| convexHull ℝ (insert 0 (Set.range fun i : Fin n => EuclideanSpace.single i (1 : ℝ))) |
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| def firstBlock (x : Euclidean 20) : Euclidean 10 := |
| WithLp.toLp 2 (fun i : Fin 10 => x (Fin.castAdd 10 i)) |
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| def secondBlock (x : Euclidean 20) : Euclidean 10 := |
| WithLp.toLp 2 (fun i : Fin 10 => x (Fin.natAdd 10 i)) |
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| def productWitness : Set (Euclidean 20) := |
| {x | firstBlock x ∈ standardSimplex 10 ∧ secondBlock x ∈ standardSimplex 10} |
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| 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 |
| sorry |
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| end Paper092 |
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| end |
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| end OAI |
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