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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 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 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 | import OAI.Geometry.ProjectionVolume.ProductFacetAreas
import OAI.Geometry.ProjectionVolume.ProjectionJacobian
namespace OAI
noncomputable section
open Set MeasureTheory
open scoped BigOperators RealInnerProductSpace ENNReal Pointwise
namespace Paper092
theorem affineHyperplane_product_area_left {r s : ℕ} (hr : 0 < r)
(v : Euclidean r) (hv : ‖v‖ = 1) (c : ℝ) (F : Set (Euclidean r))
(hF : IsCompact F) (hplane : ∀ x ∈ F, ⟪v, x⟫ = c)
(B : Set (Euclidean s)) (hB : MeasurableSet B) :
μHE[(r + s - 1 : ℕ)] (splitEuclideanProduct r s ⁻¹' (F ×ˢ B)) =
μHE[(r - 1 : ℕ)] F * volume B := by
have hv0 : v ≠ 0 := by intro h; simp [h] at hv
let H := normalHyperplane v
let T : Set (Euclidean r) := (-c) • v +ᵥ F
have hT : T ⊆ (H : Set (Euclidean r)) := by
rintro x ⟨y, hy, rfl⟩
apply Submodule.mem_orthogonal_singleton_iff_inner_right.mpr
change ⟪v, (-c) • v + y⟫ = 0
simp [inner_add_right, inner_smul_right, hv, hplane y hy]
let b := stdOrthonormalBasis ℝ H
let f : Euclidean (Module.finrank ℝ H) →ₗᵢ[ℝ] Euclidean r :=
H.subtypeₗᵢ.comp b.repr.symm.toLinearIsometry
let A : Set (Euclidean (Module.finrank ℝ H)) := f ⁻¹' T
have hTcompact : IsCompact T := hF.image (continuous_const.add continuous_id)
have hA : MeasurableSet A := hTcompact.measurableSet.preimage f.continuous.measurable
have hfimage : f '' A = T := by
apply image_preimage_eq_of_subset
intro y hy
refine ⟨b.repr ⟨y, hT hy⟩, ?_⟩
change ((b.repr.symm (b.repr ⟨y, hT hy⟩) : H) : Euclidean r) = y
rw [LinearIsometryEquiv.symm_apply_apply]
have hrank : Module.finrank ℝ H = r - 1 := normalHyperplane_finrank v hv0
have hvolA : volume A = μHE[(r - 1 : ℕ)] F := by
have h := f.toLinearMap.euclideanHausdorffMeasure_image_eq_normDet_mul_volume A
simp only [finrank_euclideanSpace_fin, LinearIsometry.normDet_eq_one,
ENNReal.ofReal_one, one_mul] at h
change μHE[Module.finrank ℝ H] (f '' A) = volume A at h
rw [hfimage] at h
conv_lhs at h => rw [hrank]
change μHE[(r - 1 : ℕ)] ((-c) • v +ᵥ F) = volume A at h
rw [measure_vadd] at h
exact h.symm
have hp := euclideanProductMap_image_area f.toLinearMap
(LinearMap.id : Euclidean s →ₗ[ℝ] Euclidean s) A B hA hB
simp only [LinearMap.id_coe, Set.image_id, LinearIsometry.normDet_eq_one,
LinearMap.normDet_id, one_mul, ENNReal.ofReal_one] at hp
change μHE[Module.finrank ℝ H + s]
(splitEuclideanProduct r s ⁻¹' ((f '' A) ×ˢ B)) = volume A * volume B at hp
rw [hfimage, hvolA] at hp
change μHE[Module.finrank ℝ H + s]
(splitEuclideanProduct r s ⁻¹' (((-c) • v +ᵥ F) ×ˢ B)) = _ at hp
rw [← zero_vadd (Euclidean s) B, splitEuclideanProduct_preimage_vadd, measure_vadd] at hp
have hdim : Module.finrank ℝ H + s = r + s - 1 := by omega
simpa only [hdim, zero_vadd] using hp
theorem affineHyperplane_product_area_right {r s : ℕ} (hr : 0 < r)
(v : Euclidean r) (hv : ‖v‖ = 1) (c : ℝ) (F : Set (Euclidean r))
(hF : IsCompact F) (hplane : ∀ x ∈ F, ⟪v, x⟫ = c)
(B : Set (Euclidean s)) (hB : MeasurableSet B) :
μHE[(s + r - 1 : ℕ)] (splitEuclideanProduct s r ⁻¹' (B ×ˢ F)) =
volume B * μHE[(r - 1 : ℕ)] F := by
have hv0 : v ≠ 0 := by intro h; simp [h] at hv
let H := normalHyperplane v
let T : Set (Euclidean r) := (-c) • v +ᵥ F
have hT : T ⊆ (H : Set (Euclidean r)) := by
rintro x ⟨y, hy, rfl⟩
apply Submodule.mem_orthogonal_singleton_iff_inner_right.mpr
change ⟪v, (-c) • v + y⟫ = 0
simp [inner_add_right, inner_smul_right, hv, hplane y hy]
let b := stdOrthonormalBasis ℝ H
let f : Euclidean (Module.finrank ℝ H) →ₗᵢ[ℝ] Euclidean r :=
H.subtypeₗᵢ.comp b.repr.symm.toLinearIsometry
let A : Set (Euclidean (Module.finrank ℝ H)) := f ⁻¹' T
have hTcompact : IsCompact T := hF.image (continuous_const.add continuous_id)
have hA : MeasurableSet A := hTcompact.measurableSet.preimage f.continuous.measurable
have hfimage : f '' A = T := by
apply image_preimage_eq_of_subset
intro y hy
refine ⟨b.repr ⟨y, hT hy⟩, ?_⟩
change ((b.repr.symm (b.repr ⟨y, hT hy⟩) : H) : Euclidean r) = y
rw [LinearIsometryEquiv.symm_apply_apply]
have hrank : Module.finrank ℝ H = r - 1 := normalHyperplane_finrank v hv0
have hvolA : volume A = μHE[(r - 1 : ℕ)] F := by
have h := f.toLinearMap.euclideanHausdorffMeasure_image_eq_normDet_mul_volume A
simp only [finrank_euclideanSpace_fin, LinearIsometry.normDet_eq_one,
ENNReal.ofReal_one, one_mul] at h
change μHE[Module.finrank ℝ H] (f '' A) = volume A at h
rw [hfimage] at h
conv_lhs at h => rw [hrank]
change μHE[(r - 1 : ℕ)] ((-c) • v +ᵥ F) = volume A at h
rw [measure_vadd] at h
exact h.symm
have hp := euclideanProductMap_image_area
(LinearMap.id : Euclidean s →ₗ[ℝ] Euclidean s) f.toLinearMap B A hB hA
simp only [LinearMap.id_coe, Set.image_id, LinearIsometry.normDet_eq_one,
LinearMap.normDet_id, one_mul, ENNReal.ofReal_one] at hp
change μHE[s + Module.finrank ℝ H]
(splitEuclideanProduct s r ⁻¹' (B ×ˢ (f '' A))) = volume B * volume A at hp
rw [hfimage, hvolA] at hp
change μHE[s + Module.finrank ℝ H]
(splitEuclideanProduct s r ⁻¹' (B ×ˢ ((-c) • v +ᵥ F))) = _ at hp
rw [← zero_vadd (Euclidean s) B, splitEuclideanProduct_preimage_vadd, measure_vadd] at hp
have hdim : s + Module.finrank ℝ H = s + r - 1 := by omega
simpa only [hdim, zero_vadd] using hp
end Paper092
end
end OAI
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