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