| import OAI.Geometry.ProjectionBody.GeometryDefinitions | |
| import Mathlib.Analysis.InnerProductSpace.Projection.Basic | |
| import Mathlib.Geometry.Euclidean.Volume.Measure | |
| namespace OAI | |
| /-! | |
| Geometric definitions for the product counterexample. | |
| The projection measure is the canonically normalized Euclidean Hausdorff | |
| measure of dimension `d - 1`, not ambient `d`-dimensional measure. | |
| -/ | |
| noncomputable section | |
| open Set MeasureTheory | |
| open scoped RealInnerProductSpace | |
| namespace ProjectionCounterexample | |
| /-- Orthogonal projection onto the hyperplane normal to `u`. -/ | |
| def project {d : ℕ} (u : E d) : E d →L[ℝ] E d := | |
| (ℝ ∙ u)ᗮ.starProjection | |
| /-- Actual (d−1)-dimensional Euclidean volume of the orthogonal projection. -/ | |
| def projectionVolume {d : ℕ} (P : Set (E d)) (u : E d) : ℝ := | |
| (Measure.euclideanHausdorffMeasure (d - 1) (project u '' P)).toReal | |
| /-- The projection body, defined entirely by its directional projection volumes. | |
| The bounds are the normalized Hausdorff measures of orthogonal projections. -/ | |
| def projectionBody {d : ℕ} (P : Set (E d)) : Set (E d) := | |
| {x | ∀ u : E d, ‖u‖ = 1 → ⟪u, x⟫ ≤ projectionVolume P u} | |
| /-- Lebesgue volume in the ambient Euclidean space. -/ | |
| def bodyVolume {d : ℕ} (P : Set (E d)) : ℝ := (volume P).toReal | |
| /-- The normalized projection-body volume. -/ | |
| def ratio {d : ℕ} (P : Set (E d)) : ℝ := | |
| bodyVolume (projectionBody P) / bodyVolume P ^ (d - 1) | |
| end ProjectionCounterexample | |
| end | |
| end OAI | |