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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 46 47 48 49 50 51 52 53 54 55 56 57 | import OAI.Geometry.ProjectionBody.NullHyperplane
import Mathlib.Analysis.InnerProductSpace.NormDet
namespace OAI
noncomputable section
open Set MeasureTheory Measure
namespace ProjectionCounterexample
variable {U V : Type*}
[NormedAddCommGroup U] [InnerProductSpace ℝ U] [FiniteDimensional ℝ U]
[MeasurableSpace U] [BorelSpace U]
[NormedAddCommGroup V] [InnerProductSpace ℝ V]
[MeasurableSpace V] [BorelSpace V]
/-- The volume of a projected affine chart, with its genuine linear Jacobian. -/
theorem measure_projected_affine_chart (L : U →ₗ[ℝ] V) (c : V)
(q : V →ₗ[ℝ] V) (s : Set U) :
μHE[Module.finrank ℝ U] (q '' ((fun x => c + L x) '' s)) =
ENNReal.ofReal (q.comp L).normDet * volume s := by
have himage : q '' ((fun x => c + L x) '' s) =
(fun y => q c + y) '' (q.comp L '' s) := by
simp only [Set.image_image]
congr 1
funext x
exact map_add q c (L x)
rw [himage, (isometry_add_left _).euclideanHausdorffMeasure_image,
(q.comp L).euclideanHausdorffMeasure_image_eq_normDet_mul_volume]
/-- Intersecting a facet chart with a genuinely transverse affine equation
has zero facet-dimensional measure, including after any linear projection. -/
theorem measure_projected_transverse_intersection (L : U →ₗ[ℝ] V) (c : V)
(q : V →ₗ[ℝ] V) (g : V →ₗ[ℝ] ℝ) (r : ℝ) (S T : Set V)
(hS : S ⊆ (fun x => c + L x) '' (univ : Set U))
(hT : T ⊆ {y | g y = r}) (hg : g.comp L ≠ 0) :
μHE[Module.finrank ℝ U] (q '' (S ∩ T)) = 0 := by
have hsub : q '' (S ∩ T) ⊆
q '' ((fun x => c + L x) '' {x | (g.comp L) x = r - g c}) := by
rintro z ⟨y, ⟨hyS, hyT⟩, rfl⟩
obtain ⟨x, _, rfl⟩ := hS hyS
refine ⟨c + L x, ⟨x, ?_, rfl⟩, rfl⟩
have heq := hT hyT
change g (c + L x) = r at heq
rw [map_add] at heq
change g (L x) = r - g c
exact eq_sub_of_add_eq' heq
apply measure_mono_null hsub
rw [measure_projected_affine_chart,
volume_linear_level_eq_zero (g.comp L) hg, mul_zero]
end ProjectionCounterexample
end
end OAI
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