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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 | import OAI.Geometry.ProjectionVolume.PrismVolume
namespace OAI
noncomputable section
open Set MeasureTheory
open scoped RealInnerProductSpace Pointwise
namespace Paper092
theorem vadd_add_set {n : ℕ} (a : Euclidean n) (A B : Set (Euclidean n)) :
(a +ᵥ A) + B = a +ᵥ (A + B) := by
ext z
constructor
· rintro ⟨x, ⟨x₀, hx, rfl⟩, y, hy, rfl⟩
exact ⟨x₀ + y, ⟨x₀, hx, y, hy, rfl⟩, (add_assoc a x₀ y).symm⟩
· rintro ⟨z, ⟨x, hx, y, hy, rfl⟩, rfl⟩
exact ⟨a + x, ⟨x, hx, rfl⟩, y, hy, add_assoc a x y⟩
theorem affineHyperplane_unitPrism_volume {n : ℕ} (hn : 0 < n)
(v : Euclidean n) (hv : ‖v‖ = 1) (c : ℝ) (F : Set (Euclidean n))
(hF : IsCompact F) (hplane : ∀ x ∈ F, ⟪v, x⟫ = c) :
volume (F + segment ℝ 0 v) = μHE[n - 1] F := by
have hv0 : v ≠ 0 := by intro h; simp [h] at hv
let H := normalHyperplane v
let T : Set (Euclidean n) := (-c) • v +ᵥ F
have hT : T ⊆ (H : Set (Euclidean n)) := 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 n :=
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 n) = y
rw [LinearIsometryEquiv.symm_apply_apply]
have hrank : Module.finrank ℝ H = n - 1 := normalHyperplane_finrank v hv0
have hvolA : volume A = μHE[n - 1] 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[n - 1] ((-c) • v +ᵥ F) = volume A at h
rw [measure_vadd] at h
exact h.symm
have hfperp (x : Euclidean (Module.finrank ℝ H)) : ⟪v, f x⟫ = 0 :=
Submodule.mem_orthogonal_singleton_iff_inner_right.mp (b.repr.symm x).property
have h := orthogonalPrism_volume f v hv hfperp (by omega) A hA
rw [hfimage, hvolA] at h
change volume (((-c) • v +ᵥ F) + segment ℝ 0 v) = _ at h
rw [vadd_add_set, measure_vadd] at h
exact h
theorem linearImage_unitPrism {n : ℕ} (L : Euclidean n ≃ₗ[ℝ] Euclidean n)
(v : Euclidean n) (F : Set (Euclidean n)) :
L '' (F + segment ℝ 0 v) = L '' F + segment ℝ 0 (L v) := by
rw [Set.image_add]
congr 1
simpa using (image_segment ℝ (L.toLinearMap.toAffineMap) (0 : Euclidean n) v)
theorem linearImage_unitPrism_volume {n : ℕ} (hn : 0 < n)
(L : Euclidean n ≃ₗ[ℝ] Euclidean n) (v : Euclidean n) (hv : ‖v‖ = 1)
(c : ℝ) (F : Set (Euclidean n)) (hF : IsCompact F)
(hplane : ∀ x ∈ F, ⟪v, x⟫ = c) :
volume (L '' F + segment ℝ 0 (L v)) = ENNReal.ofReal |L.toLinearMap.det| * μHE[n - 1] F := by
rw [← linearImage_unitPrism]
change volume (L.toLinearMap '' (F + segment ℝ 0 v)) = _
rw [volume.addHaar_image_linearMap, affineHyperplane_unitPrism_volume hn v hv c F hF hplane]
theorem linearImage_unitPrism_height {n : ℕ} (L : Euclidean n ≃ₗ[ℝ] Euclidean n)
(v : Euclidean n) (hv : ‖v‖ = 1) :
⟪linearImageNormal L v, L v⟫ = 1 / ‖L.symm.toLinearMap.adjoint v‖ := by
rw [linearImageNormal_inner, real_inner_self_eq_norm_sq, hv, one_pow]
theorem linearImage_unitPrism_offset {n : ℕ} (L : Euclidean n ≃ₗ[ℝ] Euclidean n)
(v : Euclidean n) (hv : ‖v‖ = 1) (c : ℝ) (x : Euclidean n)
(hx : ⟪v, x⟫ = c) (t : ℝ) :
⟪linearImageNormal L v, L (x + t • v)⟫ =
c / ‖L.symm.toLinearMap.adjoint v‖ + t / ‖L.symm.toLinearMap.adjoint v‖ := by
rw [linearImageNormal_inner, inner_add_right, real_inner_smul_right,
real_inner_self_eq_norm_sq, hv, one_pow, mul_one, hx, add_div]
theorem linearImage_unitPrism_volume_eq_area_mul_height {n : ℕ} (hn : 0 < n)
(L : Euclidean n ≃ₗ[ℝ] Euclidean n) (v : Euclidean n) (hv : ‖v‖ = 1)
(c : ℝ) (F : Set (Euclidean n)) (hF : IsCompact F)
(hplane : ∀ x ∈ F, ⟪v, x⟫ = c) :
(volume (L '' F + segment ℝ 0 (L v))).toReal =
(μHE[n - 1] (L '' F)).toReal * (1 / ‖L.symm.toLinearMap.adjoint v‖) := by
rw [linearImage_unitPrism_volume hn L v hv c F hF hplane, ENNReal.toReal_mul,
ENNReal.toReal_ofReal (abs_nonneg _), affineHyperplane_linearImage_area_real L v hv c F hplane]
field_simp [norm_ne_zero_iff.mpr (inverseAdjoint_ne_zero L v hv)]
end Paper092
end
end OAI
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