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2fca478 | 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 | import Mathlib.Analysis.Convex.Hull
import Mathlib.Analysis.InnerProductSpace.PiL2
namespace OAI
noncomputable section
open Set
namespace ProjectionCounterexample
abbrev E (d : ℕ) := EuclideanSpace ℝ (Fin d)
/-- The convex hull of the origin and the `d` standard unit vectors. -/
def simplex (d : ℕ) : Set (E d) :=
convexHull ℝ (insert 0 (range fun i : Fin d => EuclideanSpace.single i (1 : ℝ)))
/-- The first ten coordinates in the standard orthogonal decomposition of R²⁰. -/
def firstBlock (x : E 20) : E 10 :=
WithLp.toLp 2 fun i : Fin 10 => x (Fin.castAdd 10 i)
/-- The last ten coordinates in the standard orthogonal decomposition of R²⁰. -/
def secondBlock (x : E 20) : E 10 :=
WithLp.toLp 2 fun i : Fin 10 => x (Fin.natAdd 10 i)
/-- The genuine Cartesian product T₁₀ × T₁₀ in Euclidean R²⁰. -/
def productBody : Set (E 20) :=
{x | firstBlock x ∈ simplex 10 ∧ secondBlock x ∈ simplex 10}
/-- Compact, convex, and full-dimensional in its specified Euclidean space. -/
def IsConvexBody {d : ℕ} (P : Set (E d)) : Prop :=
IsCompact P ∧ Convex ℝ P ∧ (interior P).Nonempty
end ProjectionCounterexample
end
end OAI
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