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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 | import OAI.Geometry.ProjectionVolume.PolytopeCauchy
import OAI.Geometry.ProjectionVolume.EuclideanVolume
import OAI.Geometry.ProjectionVolume.FacetAreas
namespace OAI
open Set MeasureTheory
open scoped RealInnerProductSpace Pointwise
noncomputable section
namespace Paper092
def euclideanCube (d : ℕ) : Set (Euclidean d) :=
WithLp.toLp 2 '' Icc (0 : Fin d → ℝ) 1
theorem mem_euclideanCube {d : ℕ} (x : Euclidean d) :
x ∈ euclideanCube d ↔ (∀ i, 0 ≤ x i) ∧ ∀ i, x i ≤ 1 := by
constructor
· rintro ⟨y, hy, rfl⟩
exact hy
· intro hx
exact ⟨WithLp.ofLp x, hx, WithLp.toLp_ofLp 2 x⟩
theorem euclideanCube_isCompact (d : ℕ) : IsCompact (euclideanCube d) :=
isCompact_Icc.image (PiLp.continuous_toLp 2 _)
theorem euclideanCube_volume (d : ℕ) : volume (euclideanCube d) = 1 := by
rw [euclideanCube, volume_euclidean_image]
simp [Real.volume_Icc_pi]
def cubeNormal {d : ℕ} : Fin d ⊕ Fin d → Euclidean d
| .inl i => -EuclideanSpace.single i 1
| .inr i => EuclideanSpace.single i 1
def cubeOffset {d : ℕ} : Fin d ⊕ Fin d → ℝ
| .inl _ => 0
| .inr _ => 1
theorem cube_constraints (d : ℕ) :
{x : Euclidean d | ∀ i, ⟪cubeNormal i, x⟫ ≤ cubeOffset i} = euclideanCube d := by
ext x
simp [mem_euclideanCube, Sum.forall, cubeNormal, cubeOffset,
EuclideanSpace.inner_single_left]
def cubePolytope (d : ℕ) : HPolytope d (Fin d ⊕ Fin d) where
normal := cubeNormal
offset := cubeOffset
normal_unit i := by cases i <;> simp [cubeNormal]
compact := by rw [cube_constraints]; exact euclideanCube_isCompact d
strict_feasible := by
refine ⟨WithLp.toLp 2 (fun _ => (1 / 2 : ℝ)), ?_⟩
intro i
cases i <;> norm_num [cubeNormal, cubeOffset, EuclideanSpace.inner_single_left]
distinct := by
intro i j hij
rcases i with i | i <;> rcases j with j | j
· congr 1
by_contra h
have hc := congrArg (fun p : Euclidean d × ℝ => p.1 i) hij
simp [cubeNormal, h] at hc
· have hc := congrArg Prod.snd hij
norm_num [cubeOffset] at hc
· have hc := congrArg Prod.snd hij
norm_num [cubeOffset] at hc
· congr 1
by_contra h
have hc := congrArg (fun p : Euclidean d × ℝ => p.1 i) hij
simp [cubeNormal, h] at hc
theorem cubePolytope_body (d : ℕ) : (cubePolytope d).body = euclideanCube d :=
cube_constraints d
theorem cubePolytope_face_lower {d : ℕ} (i : Fin d) :
(cubePolytope d).face (.inl i) = {x | x ∈ euclideanCube d ∧ x i = 0} := by
ext x
change (x ∈ (cubePolytope d).body ∧ ⟪-EuclideanSpace.single i 1, x⟫ = 0) ↔ _
rw [cubePolytope_body]
simp [EuclideanSpace.inner_single_left]
theorem cubePolytope_face_upper {d : ℕ} (i : Fin d) :
(cubePolytope d).face (.inr i) = {x | x ∈ euclideanCube d ∧ x i = 1} := by
ext x
change (x ∈ (cubePolytope d).body ∧ ⟪EuclideanSpace.single i 1, x⟫ = 1) ↔ _
rw [cubePolytope_body]
simp [EuclideanSpace.inner_single_left]
theorem cube_lower_face_chart (n : ℕ) (i : Fin (n + 1)) :
facetEmbedding n i 0 '' euclideanCube n = (cubePolytope (n + 1)).face (.inl i) := by
rw [cubePolytope_face_lower]
ext y
constructor
· rintro ⟨x, hx, rfl⟩
change facetEmbedding n i 0 x ∈ euclideanCube (n + 1) ∧ facetEmbedding n i 0 x i = 0
rw [mem_euclideanCube] at hx ⊢
simp [facetEmbedding, Fin.forall_iff_succAbove i, hx]
· rintro ⟨hy, hi⟩
refine ⟨WithLp.toLp 2 (i.removeNth y), ?_, ?_⟩
· rw [mem_euclideanCube] at hy ⊢
exact ⟨fun j => hy.1 (i.succAbove j), fun j => hy.2 (i.succAbove j)⟩
· ext j
induction j using i.succAboveCases <;> simp [facetEmbedding, hi]
theorem cube_lower_face_area (n : ℕ) (i : Fin (n + 1)) :
μHE[n] ((cubePolytope (n + 1)).face (.inl i)) = 1 := by
rw [← cube_lower_face_chart]
have h := (facetEmbedding n i 0).euclideanHausdorffMeasure_image_eq_normDet_mul_volume
(euclideanCube n)
simpa only [finrank_euclideanSpace_fin, facetEmbedding_normDet_zero,
ENNReal.ofReal_one, euclideanCube_volume, mul_one] using h
theorem cube_upper_face_eq_vadd {d : ℕ} (i : Fin d) :
(cubePolytope d).face (.inr i) =
EuclideanSpace.single i (1 : ℝ) +ᵥ (cubePolytope d).face (.inl i) := by
rw [cubePolytope_face_upper, cubePolytope_face_lower]
ext y
constructor
· rintro ⟨hy, hi⟩
refine ⟨y - EuclideanSpace.single i 1, ⟨?_, ?_⟩, ?_⟩
· rw [mem_euclideanCube] at hy ⊢
constructor <;> intro j
· by_cases hj : j = i
· subst j; simp [PiLp.sub_apply, hi]
· simpa [PiLp.sub_apply, PiLp.single_apply, hj] using hy.1 j
· by_cases hj : j = i
· subst j; simp [PiLp.sub_apply, hi]
· simpa [PiLp.sub_apply, PiLp.single_apply, hj] using hy.2 j
· simp [PiLp.sub_apply, hi]
· change EuclideanSpace.single i 1 + (y - EuclideanSpace.single i 1) = y
abel
· rintro ⟨x, ⟨hx, hi⟩, rfl⟩
refine ⟨?_, ?_⟩
· rw [mem_euclideanCube] at hx ⊢
constructor <;> intro j
· by_cases hj : j = i
· subst j; simp [vadd_eq_add, PiLp.add_apply, hi]
· simpa [vadd_eq_add, PiLp.add_apply, PiLp.single_apply, hj] using hx.1 j
· by_cases hj : j = i
· subst j; simp [vadd_eq_add, PiLp.add_apply, hi]
· simpa [vadd_eq_add, PiLp.add_apply, PiLp.single_apply, hj] using hx.2 j
· simp [vadd_eq_add, PiLp.add_apply, hi]
theorem cube_upper_face_area (n : ℕ) (i : Fin (n + 1)) :
μHE[n] ((cubePolytope (n + 1)).face (.inr i)) = 1 := by
rw [cube_upper_face_eq_vadd, measure_vadd, cube_lower_face_area]
theorem cube_faceArea {d : ℕ} (hd : 0 < d) (i : Fin d ⊕ Fin d) :
(cubePolytope d).faceArea i = 1 := by
obtain ⟨n, rfl⟩ := Nat.exists_eq_succ_of_ne_zero (Nat.ne_of_gt hd)
cases i with
| inl i =>
change (μHE[n] ((cubePolytope (n + 1)).face (.inl i))).toReal = 1
rw [cube_lower_face_area, ENNReal.toReal_one]
| inr i =>
change (μHE[n] ((cubePolytope (n + 1)).face (.inr i))).toReal = 1
rw [cube_upper_face_area, ENNReal.toReal_one]
theorem euclideanCube_brightness {d : ℕ} (hd : 0 < d) (u : Euclidean d) :
brightness (euclideanCube d) u = ∑ i, |u i| := by
rw [← cubePolytope_body, (cubePolytope d).brightness_eq_sum]
simp only [cube_faceArea hd, one_mul, Fintype.sum_sum_type]
simp only [cubePolytope, cubeNormal, inner_neg_right, abs_neg,
EuclideanSpace.inner_single_right, one_mul, starRingEnd_apply, star_trivial]
ring
theorem euclideanCube_diagonal_brightness {d : ℕ} (hd : 0 < d) :
brightness (euclideanCube d) (WithLp.toLp 2 (1 : Fin d → ℝ)) = d := by
rw [euclideanCube_brightness hd]
simp
theorem euclideanCube_diagonal_extrusion {d : ℕ} (hd : 0 < d) :
(volume (euclideanCube d +
segment ℝ (0 : Euclidean d) (WithLp.toLp 2 (1 : Fin d → ℝ)))).toReal =
(volume (euclideanCube d)).toReal +
brightness (euclideanCube d) (WithLp.toLp 2 (1 : Fin d → ℝ)) := by
rw [euclideanCube_volume, ENNReal.toReal_one, euclideanCube_diagonal_brightness hd]
rw [euclideanCube, volume_euclidean_cube_add_diagonal d hd]
rw [ENNReal.toReal_add (by simp) (by simp), ENNReal.toReal_natCast, ENNReal.toReal_one]
ring
theorem euclideanCube_diagonal_projectionBody_support {d : ℕ} (hd : 0 < d) :
SupportGeometry.support (projectionBody (euclideanCube d))
(innerSL ℝ (WithLp.toLp 2 (1 : Fin d → ℝ))) = d := by
rw [← cubePolytope_body, (cubePolytope d).projectionBody_support, cubePolytope_body,
euclideanCube_diagonal_brightness hd]
end Paper092
end
end OAI
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