File size: 2,112 Bytes
465b10d | 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 | import Mathlib
namespace OAI
noncomputable section
open Set MeasureTheory
open scoped RealInnerProductSpace Pointwise
namespace Paper092
noncomputable def simplexConstant (d : ℕ) : ℝ :=
(d + 1) * (d : ℝ) ^ d / d.factorial
abbrev Euclidean (n : ℕ) := EuclideanSpace ℝ (Fin n)
def normalHyperplane {n : ℕ} (u : Euclidean n) : Submodule ℝ (Euclidean n) :=
(Submodule.span ℝ ({u} : Set (Euclidean n)))ᗮ
def projectionVolume {n : ℕ} (K : Set (Euclidean n)) (u : Euclidean n) : ENNReal :=
(volume : Measure (normalHyperplane u))
((normalHyperplane u).orthogonalProjectionOnto '' K)
def brightness {n : ℕ} (K : Set (Euclidean n)) (u : Euclidean n) : ℝ :=
‖u‖ * (projectionVolume K u).toReal
def projectionBody {n : ℕ} (K : Set (Euclidean n)) : Set (Euclidean n) :=
{y | ∀ u, ⟪u, y⟫ ≤ brightness K u}
def normalizedProjectionVolume {n : ℕ} (K : Set (Euclidean n)) : ℝ :=
(volume (projectionBody K)).toReal / (volume K).toReal ^ (n - 1)
def standardSimplex (n : ℕ) : Set (Euclidean n) :=
convexHull ℝ (insert 0 (Set.range fun i : Fin n => EuclideanSpace.single i (1 : ℝ)))
def firstBlock (x : Euclidean 20) : Euclidean 10 :=
WithLp.toLp 2 (fun i : Fin 10 => x (Fin.castAdd 10 i))
def secondBlock (x : Euclidean 20) : Euclidean 10 :=
WithLp.toLp 2 (fun i : Fin 10 => x (Fin.natAdd 10 i))
def productWitness : Set (Euclidean 20) :=
{x | firstBlock x ∈ standardSimplex 10 ∧ secondBlock x ∈ standardSimplex 10}
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
end Paper092
end
end OAI
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