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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 39 40 41 42 43 44 45 | 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
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