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695bd70 | 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 | import OAI.Geometry.RuledCovering.Frames
namespace OAI
noncomputable section
open Set MeasureTheory Filter Metric
open scoped Topology ContDiff Pointwise
namespace RuledApproximation
/-- Closed outer neighborhoods recover a closed set even when its frontier has positive measure. -/
lemma inter_reciprocal_cthickening {E : Type*} [MetricSpace E] (K : Set E)
(hK : IsClosed K) :
(⋂ n : ℕ, cthickening (1 / ((n : ℝ) + 1)) K) = K := by
apply Subset.antisymm
· intro x hx
have hxn : ∀ n : ℕ, infEDist x K ≤ ENNReal.ofReal (1 / ((n : ℝ) + 1)) := by
simpa only [mem_iInter, mem_cthickening_iff] using hx
have ht : Tendsto (fun n : ℕ => ENNReal.ofReal (1 / ((n : ℝ) + 1)))
atTop (𝓝 0) := by
simpa only [Function.comp_def, ENNReal.ofReal_zero] using ENNReal.continuous_ofReal.continuousAt.tendsto.comp
(tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ))
have hz : infEDist x K = 0 := le_antisymm (ge_of_tendsto ht (.of_forall hxn)) bot_le
exact hK.closure_subset (mem_closure_iff_infEDist_zero.mpr hz)
· exact subset_iInter fun _ => self_subset_cthickening K
/-- Small open outer neighborhoods, not regularity of the label boundary, justify the weighted
lattice count for arbitrary compact label sets. -/
theorem exists_nice_outer_integral {E : Type*} [NormedAddCommGroup E]
[NormedSpace ℝ E] [FiniteDimensional ℝ E] [MeasureSpace E] [BorelSpace E]
[IsFiniteMeasureOnCompacts (volume : Measure E)] [SFinite (volume : Measure E)]
{K U : Set E} (hK : IsCompact K) (hU : IsOpen U) (hKU : K ⊆ U)
{F : E → ℝ} (hF : Continuous F) (hFpos : ∀ x, 0 ≤ F x)
{eps : ℝ} (heps : 0 < eps) :
∃ S : Set E, IsOpen S ∧ Bornology.IsBounded S ∧ K ⊆ S ∧ S ⊆ U ∧
volume (frontier S) = 0 ∧ (∫ x in S, F x) < (∫ x in K, F x) + eps := by
let R : ℕ → ℝ := fun n => 1 / ((n : ℝ) + 1)
have hR : ∀ n, 0 < R n := fun n => by dsimp [R]; positivity
have hRlim : Tendsto R atTop (𝓝 0) := tendsto_one_div_add_atTop_nhds_zero_nat
let K' : ℕ → Set E := fun n => cthickening (R n) K
have hK' : ∀ n, IsCompact (K' n) := fun _ => hK.cthickening
have hanti : Antitone K' := by
intro n m hnm
apply cthickening_mono _ K
dsimp [R]
exact one_div_le_one_div_of_le (by positivity) (by exact_mod_cast Nat.add_le_add_right hnm 1)
have hInt : Tendsto (fun n => ∫ x in K' n, F x) atTop (𝓝 (∫ x in K, F x)) := by
have ht := tendsto_setIntegral_of_antitone (μ := volume)
(fun n => (hK' n).measurableSet) hanti
⟨0, hF.continuousOn.integrableOn_compact (μ := volume) (hK' 0)⟩
rwa [show (⋂ n, K' n) = K from inter_reciprocal_cthickening K hK.isClosed] at ht
obtain ⟨d, hd, hdU⟩ := hK.exists_cthickening_subset_open hU hKU
have hevent : ∀ᶠ n in atTop, (∫ x in K' n, F x) < (∫ x in K, F x) + eps ∧ R n < d :=
(hInt.eventually (Iio_mem_nhds (lt_add_of_pos_right _ heps))).and
(hRlim.eventually (Iio_mem_nhds hd))
obtain ⟨n, hnInt, hnd⟩ := hevent.exists
obtain ⟨r, hr, hrnull⟩ := exists_null_frontier_thickening volume K (hR n)
refine ⟨thickening r K, isOpen_thickening, hK.isBounded.thickening,
self_subset_thickening hr.1 K, ?_, hrnull, ?_⟩
· exact (thickening_subset_cthickening r K).trans
((cthickening_mono (hr.2.le.trans hnd.le) K).trans hdU)
· have hsub : thickening r K ⊆ K' n :=
(thickening_subset_cthickening r K).trans (cthickening_mono hr.2.le K)
exact (setIntegral_mono_set
(hF.continuousOn.integrableOn_compact (μ := volume) (hK' n))
(.of_forall hFpos) (.of_forall hsub)).trans_lt hnInt
/-- The square lattice in the displayed (possibly nonorthonormal) parameter coordinates. -/
def coordinateGrid (n : ℕ) : Set (Fin 2 → ℝ) :=
(n : ℝ)⁻¹ • (Submodule.span ℤ (Set.range (Pi.basisFun ℝ (Fin 2))) : Set (Fin 2 → ℝ))
lemma finite_inter_coordinateGrid {S : Set (Fin 2 → ℝ)} (hS : Bornology.IsBounded S)
{n : ℕ} (hn : n ≠ 0) : (S ∩ coordinateGrid n).Finite := by
classical
unfold coordinateGrid
rw [← Submodule.coe_pointwise_smul, ZSpan.smul _ (inv_ne_zero (Nat.cast_ne_zero.mpr hn))]
exact ZSpan.setFinite_inter _ hS
/-- A finite lattice retained around any compact label set has arbitrarily accurate weighted
cost. The finite set is the intersection with a fixed regular outer neighborhood, so no
assumption on the boundary or positive area of `K` occurs here. -/
theorem finite_weighted_grid_outer {K U : Set (Fin 2 → ℝ)}
(hK : IsCompact K) (hU : IsOpen U) (hKU : K ⊆ U)
{F : (Fin 2 → ℝ) → ℝ} (hF : Continuous F) (hFpos : ∀ x, 0 ≤ F x)
(R : ℝ) {eps : ℝ} (heps : 0 < eps) :
∀ᶠ n : ℕ in atTop, 0 < n ∧ ∃ S : Set (Fin 2 → ℝ), S.Finite ∧
S ⊆ U ∩ coordinateGrid n ∧
(∀ b ∈ coordinateGrid n, (∃ p ∈ K, dist b p ≤ R / n) → b ∈ S) ∧
(∑' b : S, F b) / (n : ℝ) ^ 2 < (∫ p in K, F p) + eps := by
obtain ⟨O, hOo, hOb, hKO, hOU, hOf, hOI⟩ :=
exists_nice_outer_integral hK hU hKU hF hFpos (half_pos heps)
obtain ⟨d, hd, hdO⟩ := hK.exists_cthickening_subset_open hOo hKO
have hsum := tendsto_tsum_div_pow_atTop_integral O F hF hOb hOo.measurableSet hOf
have hcost : ∀ᶠ n : ℕ in atTop,
(∑' b : ↑(O ∩ coordinateGrid n), F b) / (n : ℝ) ^ 2 <
(∫ p in O, F p) + eps / 2 := by
simpa only [coordinateGrid, Fintype.card_fin] using
hsum.eventually (Iio_mem_nhds (lt_add_of_pos_right _ (half_pos heps)))
have hsmall : ∀ᶠ n : ℕ in atTop, R / (n : ℝ) < d := by
have ht : Tendsto (fun n : ℕ => R / (n : ℝ)) atTop (𝓝 0) := by
simpa only [div_eq_mul_inv, mul_zero] using (tendsto_const_nhds.mul
(tendsto_inv_atTop_zero.comp tendsto_natCast_atTop_atTop) :
Tendsto (fun n : ℕ => R * (n : ℝ)⁻¹) atTop (𝓝 (R * 0)))
exact ht.eventually (Iio_mem_nhds hd)
filter_upwards [eventually_gt_atTop 0, hcost, hsmall] with n hn hcost hnsmall
refine ⟨hn, O ∩ coordinateGrid n, finite_inter_coordinateGrid hOb hn.ne',
Set.inter_subset_inter_left _ hOU, ?_, ?_⟩
· intro b hb ⟨p, hp, hbp⟩
exact ⟨hdO ((thickening_subset_cthickening d K)
(mem_thickening_iff.mpr ⟨p, hp, hbp.trans_lt hnsmall⟩)), hb⟩
· linarith
end RuledApproximation
end
end OAI
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