| 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 | |