| import OAI.Geometry.ProjectionVolume.PrismVolume |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set MeasureTheory |
| open scoped RealInnerProductSpace Pointwise |
|
|
| namespace Paper092 |
|
|
| theorem vadd_add_set (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 (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 |
| |
