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cead56f | 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 | 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
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