| 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 |
|
|
