File size: 2,874 Bytes
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 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 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 | import OAI.Geometry.ProjectionVolume.FacetLinearImage
import OAI.Geometry.ProjectionVolume.GeneralProduct
import OAI.Geometry.ProjectionVolume.SimplexPolytope
namespace OAI
universe uι uκ
noncomputable section
open Set MeasureTheory
open scoped RealInnerProductSpace
namespace Paper092
theorem affineHyperplane_linearImage_areaNormal {n : ℕ}
(L : Euclidean n ≃ₗ[ℝ] Euclidean n) (v : Euclidean n) (hv : ‖v‖ = 1)
(c : ℝ) (F : Set (Euclidean n)) (hplane : ∀ x ∈ F, ⟪v, x⟫ = c) :
(μHE[n - 1] (L '' F)).toReal • linearImageNormal L v =
(|L.toLinearMap.det| • L.symm.toLinearMap.adjoint) ((μHE[n - 1] F).toReal • v) := by
rw [affineHyperplane_linearImage_area_real L v hv c F hplane, linearImageNormal,
smul_smul, LinearMap.smul_apply, map_smul, smul_smul]
congr 1
field_simp [norm_ne_zero_iff.mpr (inverseAdjoint_ne_zero L v hv)]
theorem simplexFacet_coordinate_areaNormal (n : ℕ) (i : Fin (n + 1)) :
(μHE[n] (simplexFacet (n + 1) (some i))).toReal •
(-EuclideanSpace.single i (1 : ℝ)) =
-(1 / (n.factorial : ℝ)) • EuclideanSpace.single i (1 : ℝ) := by
rw [simplexFacet_coordinate_area, ENNReal.toReal_ofReal (by positivity)]
exact (smul_neg _ _).trans (neg_smul _ _).symm
theorem simplexFacet_diagonal_areaNormal (n : ℕ) :
(μHE[n] (simplexFacet (n + 1) none)).toReal • diagonalUnitNormal (n + 1) =
(1 / (n.factorial : ℝ)) • diagonalVector (n + 1) := by
rw [simplexFacet_diagonal_area, ENNReal.toReal_ofReal (by positivity),
diagonalUnitNormal, smul_smul]
congr 1
have hs : Real.sqrt ((n : ℝ) + 1) ≠ 0 := ne_of_gt (Real.sqrt_pos.mpr (by positivity))
push_cast
field_simp
namespace HPolytope
theorem product_areaNormal_inl {r s : ℕ} (hr : 0 < r)
{ι : Type uι} {κ : Type uκ} [Fintype ι] [Fintype κ] (A : HPolytope r ι) (B : HPolytope s κ)
(i : ι) :
splitEuclideanProduct r s ((A.product B).faceArea (.inl i) •
(A.product B).normal (.inl i)) =
((volume B.body).toReal • (A.faceArea i • A.normal i), 0) := by
rw [product_faceArea_inl hr]
change splitEuclideanProduct r s ((A.faceArea i * (volume B.body).toReal) •
productLiftLeft r s (A.normal i)) = _
simp [productLiftLeft, smul_smul, mul_comm]
theorem product_areaNormal_inr {r s : ℕ} (hs : 0 < s)
{ι : Type uι} {κ : Type uκ} [Fintype ι] [Fintype κ] (A : HPolytope r ι) (B : HPolytope s κ)
(i : κ) :
splitEuclideanProduct r s ((A.product B).faceArea (.inr i) •
(A.product B).normal (.inr i)) =
(0, (volume A.body).toReal • (B.faceArea i • B.normal i)) := by
rw [product_faceArea_inr hs]
change splitEuclideanProduct r s (((volume A.body).toReal * B.faceArea i) •
productLiftRight r s (B.normal i)) = _
simp [productLiftRight, smul_smul]
end HPolytope
end Paper092
end
end OAI
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