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465b10d | 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 46 | 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
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