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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 | import OAI.Geometry.ProjectionVolume.FiniteApproximation
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Analysis.Normed.Module.Ball.Pointwise
namespace OAI
open Set Metric Filter
open scoped Pointwise Topology
namespace Paper092
noncomputable def inscribedScale (m : ℕ) : ℝ := 1 - 1 / ((m : ℝ) + 2)
theorem inscribedScale_nonneg (m : ℕ) : 0 ≤ inscribedScale m := by
unfold inscribedScale
apply sub_nonneg.mpr
apply (div_le_one (by positivity : 0 < (m : ℝ) + 2)).mpr
linarith [Nat.cast_nonneg (α := ℝ) m]
theorem inscribedScale_lt_one (m : ℕ) : inscribedScale m < 1 := by
unfold inscribedScale
exact sub_lt_self _ (by positivity)
theorem exists_increasing_inscribed_finsets {n : ℕ} {K : Set (Euclidean n)}
(hc : IsCompact K) (hv : Convex ℝ K) (h0 : (0 : Euclidean n) ∈ interior K) :
∃ V : ℕ → Finset (Euclidean n),
Monotone (fun m => convexHull ℝ (V m : Set (Euclidean n))) ∧
(∀ m, convexHull ℝ (V m : Set (Euclidean n)) ⊆ K) ∧
(∀ m, inscribedScale m • K ⊆ interior (convexHull ℝ (V m : Set (Euclidean n)))) ∧
(∀ m, (0 : Euclidean n) ∈ interior (convexHull ℝ (V m : Set (Euclidean n)))) := by
classical
have hex := fun m => exists_inscribed_finset_sandwich hc hv h0
(inscribedScale_nonneg m) (inscribedScale_lt_one m)
choose t ht htK ht0 using hex
let V : ℕ → Finset (Euclidean n) := fun m => (Finset.range (m + 1)).biUnion t
have htV (i m : ℕ) (hi : i ≤ m) : (t i : Set (Euclidean n)) ⊆ V m := by
intro x hx
exact Finset.mem_biUnion.mpr ⟨i, Finset.mem_range.mpr (by omega), hx⟩
refine ⟨V, ?_, ?_, ?_, ?_⟩
· intro i j hij
apply convexHull_mono
intro x hx
obtain ⟨k, hk, hx⟩ := Finset.mem_biUnion.mp hx
exact Finset.mem_biUnion.mpr ⟨k, Finset.mem_range.mpr
((Finset.mem_range.mp hk).trans_le (by omega)), hx⟩
· intro m
apply convexHull_min _ hv
intro x hx
obtain ⟨i, _, hx⟩ := Finset.mem_biUnion.mp hx
exact htK i (subset_convexHull ℝ (t i : Set (Euclidean n)) hx)
· intro m
exact (ht m).trans (interior_mono (convexHull_mono (htV m m le_rfl)))
· intro m
exact interior_mono (convexHull_mono (htV m m le_rfl)) (ht0 m)
theorem add_closedBall_subset_dilate {n : ℕ} {P : Set (Euclidean n)}
(hv : Convex ℝ P) {a ε : ℝ} (ha : 0 < a) (he : 0 ≤ ε)
(hball : closedBall (0 : Euclidean n) a ⊆ P) :
P + closedBall (0 : Euclidean n) ε ⊆ (1 + ε / a) • P := by
have heq : (ε / a) • closedBall (0 : Euclidean n) a = closedBall 0 ε := by
rw [smul_closedBall _ _ ha.le, smul_zero, Real.norm_eq_abs,
abs_of_nonneg (div_nonneg he ha.le), div_mul_cancel₀ _ ha.ne']
rw [← heq, hv.add_smul zero_le_one (div_nonneg he ha.le), one_smul]
exact add_subset_add subset_rfl (image_mono hball)
theorem inscribed_thickening {n : ℕ} {K P : Set (Euclidean n)} {R : ℝ}
(hR : ∀ x ∈ K, ‖x‖ ≤ R) (m : ℕ) (hP : inscribedScale m • K ⊆ P) :
K ⊆ P + closedBall (0 : Euclidean n) (R / ((m : ℝ) + 2)) := by
intro x hx
refine ⟨inscribedScale m • x, hP ⟨x, hx, rfl⟩,
(1 / ((m : ℝ) + 2)) • x, ?_, ?_⟩
· simp only [mem_closedBall, dist_zero_right, norm_smul, Real.norm_eq_abs,
abs_of_nonneg (by positivity : 0 ≤ 1 / ((m : ℝ) + 2))]
calc
1 / ((m : ℝ) + 2) * ‖x‖ ≤ 1 / ((m : ℝ) + 2) * R :=
mul_le_mul_of_nonneg_left (hR x hx) (by positivity)
_ = R / ((m : ℝ) + 2) := by ring
· change inscribedScale m • x + (1 / ((m : ℝ) + 2)) • x = x
rw [← add_smul]
simp [inscribedScale]
theorem tendsto_approximation_error (R : ℝ) :
Tendsto (fun m : ℕ => R / ((m : ℝ) + 2)) atTop (𝓝 0) := by
simpa only [Function.comp_def, Nat.cast_add, Nat.cast_ofNat] using
(tendsto_const_div_atTop_nhds_zero_nat R).comp (tendsto_add_atTop_nat 2)
theorem exists_inscribed_polytope_sequence {n : ℕ} {K : Set (Euclidean n)}
(hc : IsCompact K) (hv : Convex ℝ K) (h0 : (0 : Euclidean n) ∈ interior K) :
∃ (V : ℕ → Finset (Euclidean n)) (a R : ℝ),
0 < a ∧ 0 < R ∧ K ⊆ closedBall 0 R ∧
Monotone (fun m => convexHull ℝ (V m : Set (Euclidean n))) ∧
(∀ m, closedBall (0 : Euclidean n) a ⊆ convexHull ℝ (V m : Set (Euclidean n))) ∧
(∀ m, convexHull ℝ (V m : Set (Euclidean n)) ⊆ K ∧
K ⊆ convexHull ℝ (V m : Set (Euclidean n)) + closedBall 0 (R / ((m : ℝ) + 2)) ∧
convexHull ℝ (V m : Set (Euclidean n)) + closedBall 0 (R / ((m : ℝ) + 2)) ⊆
(1 + (R / ((m : ℝ) + 2)) / a) • convexHull ℝ (V m : Set (Euclidean n))) ∧
Tendsto (fun m : ℕ => R / ((m : ℝ) + 2)) atTop (𝓝 0) := by
obtain ⟨V, hmono, hVK, hscale, hzero⟩ := exists_increasing_inscribed_finsets hc hv h0
obtain ⟨r, hr, hball⟩ := Metric.mem_nhds_iff.mp (mem_interior_iff_mem_nhds.mp (hzero 0))
obtain ⟨R, hRpos, hR⟩ := hc.isBounded.exists_pos_norm_le
have hball0 : closedBall (0 : Euclidean n) (r / 2) ⊆ convexHull ℝ (V 0 : Set (Euclidean n)) :=
(closedBall_subset_ball (by linarith)).trans hball
have hballm : ∀ m, closedBall (0 : Euclidean n) (r / 2) ⊆
convexHull ℝ (V m : Set (Euclidean n)) :=
fun m => hball0.trans (hmono (Nat.zero_le m))
refine ⟨V, r / 2, R, half_pos hr, hRpos, ?_, hmono, hballm, ?_,
tendsto_approximation_error R⟩
· intro x hx
simpa only [mem_closedBall, dist_zero_right] using hR x hx
· intro m
exact ⟨hVK m, inscribed_thickening hR m ((hscale m).trans interior_subset),
add_closedBall_subset_dilate (convex_convexHull ℝ _) (half_pos hr) (by positivity) (hballm m)⟩
end Paper092
end OAI
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