| import OAI.Geometry.ProjectionVolume.Model | |
| namespace OAI | |
| noncomputable section | |
| open Set MeasureTheory | |
| open scoped RealInnerProductSpace | |
| namespace Paper092 | |
| theorem normalHyperplane_volume_is_canonical {n : ℕ} (u : Euclidean n) : | |
| (volume : Measure (normalHyperplane u)) = | |
| (stdOrthonormalBasis ℝ (normalHyperplane u)).toBasis.addHaar := rfl | |
| theorem normalHyperplane_finrank {n : ℕ} (u : Euclidean n) (hu : u ≠ 0) : | |
| Module.finrank ℝ (normalHyperplane u) = n - 1 := by | |
| have h := (Submodule.span ℝ ({u} : Set (Euclidean n))).finrank_add_finrank_orthogonal | |
| simp only [finrank_span_singleton hu, finrank_euclideanSpace_fin] at h | |
| change Module.finrank ℝ ((Submodule.span ℝ ({u} : Set (Euclidean n)))ᗮ) = n - 1 | |
| omega | |
| end Paper092 | |
| end | |
| end OAI | |