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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 | import OAI.Geometry.ProjectionVolume.ProductReconstruction
import OAI.Geometry.ProjectionVolume.FacetLinearImage
import OAI.Geometry.ProjectionVolume.SimplexCoordinates
import OAI.Geometry.ProjectionVolume.SimplexVolume
namespace OAI
noncomputable section
open Set MeasureTheory
open scoped BigOperators RealInnerProductSpace Pointwise
namespace Paper092
def orthogonalPrismMap {m n : ℕ} (f : Euclidean m →ₗᵢ[ℝ] Euclidean n)
(v : Euclidean n) : Euclidean (m + 1) →ₗ[ℝ] Euclidean n where
toFun x := f (splitEuclideanProduct m 1 x).1 + ((splitEuclideanProduct m 1 x).2 0) • v
map_add' x y := by simp only [map_add, Prod.fst_add, Prod.snd_add, PiLp.add_apply, add_smul]; abel
map_smul' t x := by
simp only [map_smul, Prod.smul_fst, Prod.smul_snd, PiLp.smul_apply, smul_eq_mul, smul_add,
smul_smul, RingHom.id_apply]
theorem orthogonalPrismMap_inner {m n : ℕ} (f : Euclidean m →ₗᵢ[ℝ] Euclidean n)
(v : Euclidean n) (hv : ‖v‖ = 1) (hf : ∀ x, ⟪v, f x⟫ = 0)
(x y : Euclidean (m + 1)) :
⟪orthogonalPrismMap f v x, orthogonalPrismMap f v y⟫ = ⟪x, y⟫ := by
have hfv (a : Euclidean m) : ⟪f a, v⟫ = 0 := by rw [real_inner_comm, hf]
have hone (a b : Euclidean 1) : ⟪a, b⟫ = a 0 * b 0 := by
simp [EuclideanSpace.inner_eq_star_dotProduct, dotProduct, mul_comm]
rw [splitEuclideanProduct_inner m 1 x y]
change ⟪f (splitEuclideanProduct m 1 x).1 + ((splitEuclideanProduct m 1 x).2 0) • v,
f (splitEuclideanProduct m 1 y).1 + ((splitEuclideanProduct m 1 y).2 0) • v⟫ = _
simp only [inner_add_left, inner_add_right, real_inner_smul_left, real_inner_smul_right,
f.inner_map_map, hf, hfv, real_inner_self_eq_norm_sq, hv, one_pow, mul_zero, zero_add,
add_zero, mul_one, hone]
ring
def orthogonalPrismEmbedding {m n : ℕ} (f : Euclidean m →ₗᵢ[ℝ] Euclidean n)
(v : Euclidean n) (hv : ‖v‖ = 1) (hf : ∀ x, ⟪v, f x⟫ = 0) :
Euclidean (m + 1) →ₗᵢ[ℝ] Euclidean n where
toLinearMap := orthogonalPrismMap f v
norm_map' x := by
apply (sq_eq_sq₀ (norm_nonneg _) (norm_nonneg _)).mp
rw [← real_inner_self_eq_norm_sq, ← real_inner_self_eq_norm_sq]
exact orthogonalPrismMap_inner f v hv hf x x
theorem mem_standardSimplex_one (x : Euclidean 1) :
x ∈ standardSimplex 1 ↔ 0 ≤ x 0 ∧ x 0 ≤ 1 := by
simp [mem_standardSimplex_iff, Fin.forall_fin_one]
theorem orthogonalPrismMap_image {m n : ℕ} (f : Euclidean m →ₗᵢ[ℝ] Euclidean n)
(v : Euclidean n) (A : Set (Euclidean m)) :
orthogonalPrismMap f v '' cartesianBody A (standardSimplex 1) =
f '' A + segment ℝ 0 v := by
ext z
constructor
· rintro ⟨x, hx, rfl⟩
refine ⟨f (splitEuclideanProduct m 1 x).1, ⟨_, hx.1, rfl⟩,
((splitEuclideanProduct m 1 x).2 0) • v, ?_, rfl⟩
rw [segment_eq_image]
refine ⟨(splitEuclideanProduct m 1 x).2 0, (mem_standardSimplex_one _).mp hx.2, ?_⟩
simp
· rintro ⟨a, ⟨x, hx, rfl⟩, b, hb, rfl⟩
rw [segment_eq_image] at hb
obtain ⟨t, ht, rfl⟩ := hb
let y : Euclidean 1 := WithLp.toLp 2 (fun _ => t)
refine ⟨(splitEuclideanProduct m 1).symm (x, y), ?_, ?_⟩
· rw [cartesianBody_eq_preimage, Set.mem_preimage, ContinuousLinearEquiv.apply_symm_apply]
exact ⟨hx, (mem_standardSimplex_one y).mpr ht⟩
· change f (splitEuclideanProduct m 1 ((splitEuclideanProduct m 1).symm (x, y))).1 +
((splitEuclideanProduct m 1 ((splitEuclideanProduct m 1).symm (x, y))).2 0) • v = _
simp [y]
theorem orthogonalPrism_volume {m n : ℕ} (f : Euclidean m →ₗᵢ[ℝ] Euclidean n)
(v : Euclidean n) (hv : ‖v‖ = 1) (hf : ∀ x, ⟪v, f x⟫ = 0) (hdim : m + 1 = n)
(A : Set (Euclidean m)) (hA : MeasurableSet A) :
volume (f '' A + segment ℝ 0 v) = volume A := by
let g := orthogonalPrismEmbedding f v hv hf
have h := g.toLinearMap.euclideanHausdorffMeasure_image_eq_normDet_mul_volume
(cartesianBody A (standardSimplex 1))
simp only [finrank_euclideanSpace_fin, LinearIsometry.normDet_eq_one,
ENNReal.ofReal_one, one_mul] at h
change μHE[m + 1] (orthogonalPrismMap f v '' cartesianBody A (standardSimplex 1)) = _ at h
rw [orthogonalPrismMap_image, cartesianBody_volume _ _ hA
(standardSimplex_isCompact 1).measurableSet, standardSimplex_volume] at h
norm_num at h
rw [hdim, EuclideanSpace.euclideanHausdorffMeasure_eq_volume] at h
exact h
end Paper092
end
end OAI
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