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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 | import OAI.Geometry.ProjectionVolume.Brightness
import Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls
import Mathlib.Analysis.Normed.Module.Ball.Pointwise
namespace OAI
open Set Metric MeasureTheory
open scoped Pointwise RealInnerProductSpace
namespace Paper092
noncomputable def euclideanUnitBallVolume (d : ℕ) : ℝ :=
(volume (closedBall (0 : Euclidean d) 1)).toReal
theorem projection_image_closedBall {d : ℕ} (u : Euclidean d) (R : ℝ) :
(normalHyperplane u).orthogonalProjectionOnto '' closedBall (0 : Euclidean d) R =
closedBall (0 : normalHyperplane u) R := by
ext y
constructor
· rintro ⟨x, hx, rfl⟩
apply mem_closedBall_zero_iff.mpr
exact ((normalHyperplane u).norm_orthogonalProjectionOnto_apply_le x).trans
(mem_closedBall_zero_iff.mp hx)
· intro hy
refine ⟨y.val, ?_, (normalHyperplane u).orthogonalProjectionOnto_mem_subspace_eq_self y⟩
simpa only [mem_closedBall, dist_zero_right, Submodule.norm_coe] using hy
theorem projectionVolume_closedBall_real {d : ℕ} (hd : 2 ≤ d)
(u : Euclidean d) (hu : u ≠ 0) (R : ℝ) (hR : 0 ≤ R) :
(projectionVolume (closedBall (0 : Euclidean d) R) u).toReal =
euclideanUnitBallVolume (d - 1) * R ^ (d - 1) := by
let : Nontrivial (normalHyperplane u) := Module.nontrivial_of_finrank_pos (by
rw [normalHyperplane_finrank u hu]
omega)
let : Nontrivial (Euclidean (d - 1)) := Module.nontrivial_of_finrank_pos (by
rw [finrank_euclideanSpace_fin]
omega)
have hunit : (volume (closedBall (0 : normalHyperplane u) 1)) =
volume (closedBall (0 : Euclidean (d - 1)) 1) := by
rw [InnerProductSpace.volume_closedBall, InnerProductSpace.volume_closedBall,
normalHyperplane_finrank u hu, finrank_euclideanSpace_fin]
have hball : R • closedBall (0 : normalHyperplane u) 1 = closedBall 0 R := by
rw [_root_.smul_closedBall _ _ zero_le_one, smul_zero, Real.norm_eq_abs, abs_of_nonneg hR, mul_one]
unfold projectionVolume
rw [projection_image_closedBall, ← hball, Measure.addHaar_smul,
normalHyperplane_finrank u hu, abs_pow, abs_of_nonneg hR,
ENNReal.toReal_mul, ENNReal.toReal_ofReal (pow_nonneg hR _), hunit]
exact mul_comm _ _
theorem brightness_closedBall {d : ℕ} (hd : 2 ≤ d) (R : ℝ) (hR : 0 ≤ R)
(u : Euclidean d) :
brightness (closedBall (0 : Euclidean d) R) u =
‖u‖ * (euclideanUnitBallVolume (d - 1) * R ^ (d - 1)) := by
by_cases hu : u = 0
· simp [hu, brightness]
· unfold brightness
rw [projectionVolume_closedBall_real hd u hu R hR]
theorem brightness_le_ball_bound {d : ℕ} (hd : 2 ≤ d) {K : Set (Euclidean d)}
{R : ℝ} (hR : 0 ≤ R) (hK : K ⊆ closedBall 0 R) (u : Euclidean d) :
brightness K u ≤ ‖u‖ * (euclideanUnitBallVolume (d - 1) * R ^ (d - 1)) := by
exact (brightness_mono hK (isCompact_closedBall _ _) u).trans_eq
(brightness_closedBall hd R hR u)
theorem ball_subset_projectionBody {d : ℕ} (hd : 2 ≤ d) {K : Set (Euclidean d)}
(hK : IsCompact K) {a : ℝ} (ha : 0 ≤ a) (hball : closedBall 0 a ⊆ K) :
closedBall (0 : Euclidean d) (euclideanUnitBallVolume (d - 1) * a ^ (d - 1)) ⊆
projectionBody K := by
intro y hy u
calc
⟪u, y⟫ ≤ ‖u‖ * ‖y‖ := real_inner_le_norm _ _
_ ≤ ‖u‖ * (euclideanUnitBallVolume (d - 1) * a ^ (d - 1)) :=
mul_le_mul_of_nonneg_left (mem_closedBall_zero_iff.mp hy) (norm_nonneg u)
_ = brightness (closedBall (0 : Euclidean d) a) u := (brightness_closedBall hd a ha u).symm
_ ≤ brightness K u := brightness_mono hball hK u
theorem projectionBody_subset_ball {d : ℕ} (hd : 2 ≤ d) {K : Set (Euclidean d)}
{R : ℝ} (hR : 0 ≤ R) (hK : K ⊆ closedBall 0 R) :
projectionBody K ⊆
closedBall (0 : Euclidean d) (euclideanUnitBallVolume (d - 1) * R ^ (d - 1)) := by
intro y hy
have hc : 0 ≤ euclideanUnitBallVolume (d - 1) * R ^ (d - 1) := by
exact mul_nonneg ENNReal.toReal_nonneg (pow_nonneg hR _)
have h := (hy y).trans (brightness_le_ball_bound hd hR hK y)
rw [real_inner_self_eq_norm_sq] at h
apply mem_closedBall_zero_iff.mpr
nlinarith [norm_nonneg y]
end Paper092
end OAI
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