| import Mathlib.Analysis.Convex.Hull |
| import Mathlib.Analysis.InnerProductSpace.PiL2 |
| import Mathlib.Analysis.InnerProductSpace.Projection.Basic |
| import Mathlib.Geometry.Euclidean.Volume.Measure |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| open Set MeasureTheory |
| open scoped RealInnerProductSpace |
|
|
| namespace ProjectionCounterexample |
|
|
| abbrev E (d : ℕ) := EuclideanSpace ℝ (Fin d) |
|
|
| def simplex (d : ℕ) : Set (E d) := |
| convexHull ℝ (insert 0 (range fun i : Fin d => EuclideanSpace.single i (1 : ℝ))) |
|
|
| def IsConvexBody {d : ℕ} (P : Set (E d)) : Prop := |
| IsCompact P ∧ Convex ℝ P ∧ (interior P).Nonempty |
|
|
| def project {d : ℕ} (u : E d) : E d →L[ℝ] E d := |
| (ℝ ∙ u)ᗮ.starProjection |
|
|
| def projectionVolume {d : ℕ} (P : Set (E d)) (u : E d) : ℝ := |
| (Measure.euclideanHausdorffMeasure (d - 1) (project u '' P)).toReal |
|
|
| def projectionBody {d : ℕ} (P : Set (E d)) : Set (E d) := |
| {x | ∀ u : E d, ‖u‖ = 1 → ⟪u, x⟫ ≤ projectionVolume P u} |
|
|
| def bodyVolume {d : ℕ} (P : Set (E d)) : ℝ := (volume P).toReal |
|
|
| def ratio {d : ℕ} (P : Set (E d)) : ℝ := |
| bodyVolume (projectionBody P) / bodyVolume P ^ (d - 1) |
|
|
| theorem universal_simplex_upper_bound_false : |
| ¬ (∀ P : Set (E 20), IsConvexBody P → ratio P ≤ ratio (simplex 20)) := by |
| sorry |
|
|
| end ProjectionCounterexample |
|
|
| end |
|
|
| end OAI |
|
|
