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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 | import OAI.Geometry.ProjectionVolume.BallBrightness
import OAI.Geometry.ProjectionVolume.SupportLimit
import OAI.Geometry.ProjectionVolume.InnerSequence
namespace OAI
open Set Metric Filter MeasureTheory
open scoped Pointwise Topology RealInnerProductSpace
namespace Paper092
theorem volume_smul_real {d : ℕ} (P : Set (Euclidean d)) {c : ℝ} (hc : 0 ≤ c) :
(volume (c • P)).toReal = c ^ d * (volume P).toReal := by
rw [Measure.addHaar_smul, finrank_euclideanSpace_fin, abs_pow, abs_of_nonneg hc,
ENNReal.toReal_mul, ENNReal.toReal_ofReal (pow_nonneg hc _)]
theorem volume_real_mono {d : ℕ} {P K : Set (Euclidean d)} (hK : IsCompact K) (h : P ⊆ K) :
(volume P).toReal ≤ (volume K).toReal :=
ENNReal.toReal_mono hK.measure_lt_top.ne (measure_mono h)
theorem approximation_brightness_bound {d : ℕ} (hd : 2 ≤ d)
{P K : Set (Euclidean d)} (hP : IsCompact P) (hK : IsCompact K)
{scale R : ℝ} (hscale : 1 ≤ scale) (hR : 0 ≤ R) (hPK : P ⊆ K) (hKP : K ⊆ scale • P)
(hKR : K ⊆ closedBall 0 R) (u : Euclidean d) (hu : ‖u‖ = 1) :
0 ≤ brightness K u - brightness P u ∧
brightness K u - brightness P u ≤
(scale ^ (d - 1) - 1) * (euclideanUnitBallVolume (d - 1) * R ^ (d - 1)) := by
have hmono := brightness_mono hPK hK u
have hupper : brightness K u ≤ scale ^ (d - 1) * brightness P u := by
have hh := brightness_mono hKP (hP.smul scale) u
simpa only [brightness_body_smul, abs_of_nonneg (zero_le_one.trans hscale)] using hh
have hbound : brightness P u ≤ euclideanUnitBallVolume (d - 1) * R ^ (d - 1) :=
hmono.trans (by simpa only [hu, one_mul] using brightness_le_ball_bound hd hR hKR u)
have hpow : 1 ≤ scale ^ (d - 1) := one_le_pow₀ hscale
refine ⟨sub_nonneg.mpr hmono, ?_⟩
calc
brightness K u - brightness P u ≤ (scale ^ (d - 1) - 1) * brightness P u := by nlinarith
_ ≤ _ := mul_le_mul_of_nonneg_left hbound (sub_nonneg.mpr hpow)
theorem approximation_projectionBody_sandwich {d : ℕ} {P K : Set (Euclidean d)}
(hP : IsCompact P) (hK : IsCompact K) {scale : ℝ} (hscale : 0 < scale)
(hPK : P ⊆ K) (hKP : K ⊆ scale • P) :
projectionBody P ⊆ projectionBody K ∧
projectionBody K ⊆ scale ^ (d - 1) • projectionBody P := by
refine ⟨projectionBody_mono hPK hK, ?_⟩
have hh := projectionBody_mono hKP (hP.smul scale)
simpa only [projectionBody_body_smul P scale hscale.ne', abs_of_pos hscale] using hh
theorem approximation_volume_sandwich {d : ℕ} {P K : Set (Euclidean d)}
(hP : IsCompact P) (hK : IsCompact K) {scale : ℝ} (hscale : 0 ≤ scale)
(hPK : P ⊆ K) (hKP : K ⊆ scale • P) :
(volume P).toReal ≤ (volume K).toReal ∧
(volume K).toReal ≤ scale ^ d * (volume P).toReal := by
refine ⟨volume_real_mono hK hPK, ?_⟩
exact (volume_real_mono (hP.smul scale) hKP).trans_eq (volume_smul_real P hscale)
theorem approximation_projectionBody_volume_sandwich {d : ℕ} {P K : Set (Euclidean d)}
(hP : IsCompact P) (hK : IsCompact K) {scale : ℝ} (hscale : 0 < scale)
(hPK : P ⊆ K) (hKP : K ⊆ scale • P) :
(volume (projectionBody P)).toReal ≤ (volume (projectionBody K)).toReal ∧
(volume (projectionBody K)).toReal ≤
scale ^ (d * (d - 1)) * (volume (projectionBody P)).toReal := by
obtain ⟨hPiPK, hPiKP⟩ := approximation_projectionBody_sandwich hP hK hscale hPK hKP
have hh := approximation_volume_sandwich (projectionBody_isCompact P)
(projectionBody_isCompact K) (pow_nonneg hscale.le _) hPiPK hPiKP
simpa only [← pow_mul, Nat.mul_comm (d - 1) d] using hh
theorem approximation_brightness_uniform {d : ℕ} (hd : 2 ≤ d)
{P : ℕ → Set (Euclidean d)} {K : Set (Euclidean d)}
(hP : ∀ m, IsCompact (P m)) (hK : IsCompact K)
{scale : ℕ → ℝ} (hscale : ∀ m, 1 ≤ scale m) (hscalelim : Tendsto scale atTop (𝓝 1))
{R : ℝ} (hR : 0 ≤ R) (hKR : K ⊆ closedBall 0 R)
(hPK : ∀ m, P m ⊆ K) (hKP : ∀ m, K ⊆ scale m • P m) :
TendstoUniformlyOn (fun m u => brightness (P m) u) (brightness K) atTop (sphere 0 1) := by
let C := euclideanUnitBallVolume (d - 1) * R ^ (d - 1)
have hcont : Continuous (fun r : ℝ => (r ^ (d - 1) - 1) * C) := by fun_prop
have hlim : Tendsto (fun m => (scale m ^ (d - 1) - 1) * C) atTop (𝓝 0) := by
simpa only [Function.comp_def, one_pow, sub_self, zero_mul] using (hcont.tendsto 1).comp hscalelim
apply Metric.tendstoUniformlyOn_iff.mpr
intro ε hε
filter_upwards [hlim.eventually (Iio_mem_nhds hε)] with m hm
intro u hu
have hunit : ‖u‖ = 1 := mem_sphere_zero_iff_norm.mp hu
have hb := approximation_brightness_bound hd (hP m) hK (hscale m) hR (hPK m) (hKP m) hKR u hunit
rw [Real.dist_eq, abs_of_nonneg hb.1]
exact hb.2.trans_lt hm
theorem tendsto_of_multiplicative_sandwich {f q : ℕ → ℝ} {L : ℝ}
(hq : ∀ m, 0 < q m) (hqlim : Tendsto q atTop (𝓝 1))
(hupper : ∀ m, f m ≤ L) (hlower : ∀ m, L ≤ q m * f m) :
Tendsto f atTop (𝓝 L) := by
have hlim : Tendsto (fun m => L / q m) atTop (𝓝 L) := by
have hcont : ContinuousAt (fun x : ℝ => L / x) 1 := by fun_prop (disch := norm_num)
simpa only [Function.comp_def, div_one] using hcont.tendsto.comp hqlim
apply tendsto_of_tendsto_of_tendsto_of_le_of_le hlim tendsto_const_nhds
· intro m
exact (div_le_iff₀ (hq m)).mpr (by simpa only [mul_comm] using hlower m)
· exact hupper
theorem approximation_volume_tendsto {d : ℕ}
{P : ℕ → Set (Euclidean d)} {K : Set (Euclidean d)}
(hP : ∀ m, IsCompact (P m)) (hK : IsCompact K)
{scale : ℕ → ℝ} (hscale : ∀ m, 0 < scale m) (hscalelim : Tendsto scale atTop (𝓝 1))
(hPK : ∀ m, P m ⊆ K) (hKP : ∀ m, K ⊆ scale m • P m) :
Tendsto (fun m => (volume (P m)).toReal) atTop (𝓝 (volume K).toReal) ∧
Tendsto (fun m => (volume (projectionBody (P m))).toReal) atTop
(𝓝 (volume (projectionBody K)).toReal) := by
have hq (k : ℕ) : Tendsto (fun m => scale m ^ k) atTop (𝓝 1) := by
have hcont : Continuous (fun x : ℝ => x ^ k) := by fun_prop
simpa only [Function.comp_def, one_pow] using (hcont.tendsto 1).comp hscalelim
constructor
· apply tendsto_of_multiplicative_sandwich (fun m => pow_pos (hscale m) d) (hq d)
· exact fun m => (approximation_volume_sandwich (hP m) hK (hscale m).le (hPK m) (hKP m)).1
· exact fun m => (approximation_volume_sandwich (hP m) hK (hscale m).le (hPK m) (hKP m)).2
· apply tendsto_of_multiplicative_sandwich (fun m => pow_pos (hscale m) (d * (d - 1)))
(hq (d * (d - 1)))
· exact fun m => (approximation_projectionBody_volume_sandwich (hP m) hK (hscale m) (hPK m) (hKP m)).1
· exact fun m => (approximation_projectionBody_volume_sandwich (hP m) hK (hscale m) (hPK m) (hKP m)).2
end Paper092
end OAI
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