| import Mathlib |
| import OAI.Geometry.IntegralFillings.Model |
|
|
| namespace OAI |
|
|
| section |
| open Filter Set |
| open Set Filter MeasureTheory TopologicalSpace |
| open scoped Topology ENNReal |
| open Set MeasureTheory |
| open scoped RealInnerProductSpace |
| open Matrix |
| open scoped RealInnerProductSpace MatrixOrder |
| open Set Filter MeasureTheory |
| open scoped Topology ENNReal NNReal |
| open MeasureTheory Filter Set Metric |
| open scoped Topology Pointwise NNReal |
| open Set MeasureTheory Measure Filter Module |
| open Set Filter MeasureTheory Measure ContinuousLinearMap |
| open scoped Topology Convolution NNReal |
| open Set Filter MeasureTheory Measure Metric |
| open scoped Topology ContDiff |
| open Set Filter Metric |
| open scoped Topology NNReal |
|
|
| namespace SharpIntegralFillings |
| variable {E : Type*} [PseudoMetricSpace E] |
| lemma lipschitzOn_limit {s : Set E} {K : ℝ≥0} {fs : ℕ → E → ℝ} {f : E → ℝ} |
| (hs : ∀ j, LipschitzOnWith K (fs j) s) |
| (hl : ∀ x ∈ s, Tendsto (fun j => fs j x) atTop (𝓝 (f x))) : |
| LipschitzOnWith K f s := by |
| apply LipschitzOnWith.of_dist_le_mul |
| intro x hx y hy |
| exact le_of_tendsto ((hl x hx).dist (hl y hy)) |
| (Eventually.of_forall fun j => (hs j).dist_le_mul x hx y hy) |
|
|
| lemma uniform_limit_on_totallyBounded {s : Set E} (hs : TotallyBounded s) |
| {K : ℝ≥0} {fs : ℕ → E → ℝ} {f : E → ℝ} |
| (hfs : ∀ j, LipschitzOnWith K (fs j) s) |
| (hl : ∀ x ∈ s, Tendsto (fun j => fs j x) atTop (𝓝 (f x))) : |
| TendstoUniformlyOn fs f atTop s := by |
| have hf := lipschitzOn_limit hfs hl |
| apply Metric.tendstoUniformlyOn_iff.mpr |
| intro ε hε |
| let δ := ε / (4 * ((K : ℝ)+1)) |
| have hδ : 0 < δ := by dsimp [δ]; positivity |
| have hδeq : 4 * ((K : ℝ)+1) * δ = ε := by dsimp [δ]; field_simp |
| obtain ⟨t, hts, ht, hcov⟩ := finite_approx_of_totallyBounded hs δ hδ |
| have hpt : ∀ᶠ j in atTop, ∀ y ∈ t, dist (f y) (fs j y) < ε / 2 := by |
| apply (ht.eventually_all.mpr) |
| intro y hy |
| have hh := (Metric.tendsto_atTop.mp (hl y (hts hy))) (ε/2) (half_pos hε) |
| exact (eventually_atTop.mpr hh).mono fun j hj => by simpa only [dist_comm] using hj |
| filter_upwards [hpt] with j hj x hx |
| obtain ⟨y, hyt, hxy⟩ := mem_iUnion₂.mp (hcov hx) |
| have hd : dist x y < δ := hxy |
| have H := dist_triangle4 (f x) (f y) (fs j y) (fs j x) |
| have H₁ := hf.dist_le_mul x hx y (hts hyt) |
| have H₂ := (hfs j).dist_le_mul y (hts hyt) x hx |
| rw [dist_comm y x] at H₂ |
| have H₃ := hj y hyt |
| have H₄ : (K : ℝ) * dist x y ≤ (K : ℝ) * δ := |
| mul_le_mul_of_nonneg_left hd.le K.coe_nonneg |
| nlinarith [K.coe_nonneg] |
|
|
| noncomputable def mcShane (s : Set E) (K : ℝ≥0) (f : E → ℝ) (x : E) : ℝ := |
| ⨅ z : s, f z + K * dist x z |
|
|
| lemma mcShane_bddBelow {s : Set E} (hs : s.Nonempty) {K : ℝ≥0} {f : E → ℝ} |
| (hf : LipschitzOnWith K f s) (x : E) : |
| BddBelow (range fun z : s => f z + K * dist x z) := by |
| obtain ⟨z,hz⟩ := hs |
| refine ⟨f z - K * dist x z, ?_⟩ |
| rintro w ⟨t,rfl⟩ |
| dsimp |
| rw [_root_.sub_le_iff_le_add, add_assoc, ← mul_add, add_comm (dist x t)] |
| calc |
| f z ≤ f t + K * dist z t := hf.le_add_mul hz t.2 |
| _ ≤ f t + K * (dist x z + dist x t) := by gcongr; apply dist_triangle_left |
|
|
| lemma mcShane_eqOn {s : Set E} (hs : s.Nonempty) {K : ℝ≥0} {f : E → ℝ} |
| (hf : LipschitzOnWith K f s) : EqOn f (mcShane s K f) s := by |
| let : Nonempty s := hs.to_subtype |
| intro x hx |
| refine le_antisymm (le_ciInf fun y => hf.le_add_mul hx y.2) ?_ |
| simpa only [mcShane, add_zero, Subtype.coe_mk, mul_zero, dist_self] using |
| ciInf_le (mcShane_bddBelow hs hf x) ⟨x,hx⟩ |
|
|
| lemma mcShane_lipschitz {s : Set E} (hs : s.Nonempty) {K : ℝ≥0} {f : E → ℝ} |
| (hf : LipschitzOnWith K f s) : LipschitzWith K (mcShane s K f) := by |
| let : Nonempty s := hs.to_subtype |
| apply LipschitzWith.of_le_add_mul |
| intro x y |
| rw [← _root_.sub_le_iff_le_add] |
| refine le_ciInf fun z => ?_ |
| rw [_root_.sub_le_iff_le_add] |
| calc |
| mcShane s K f x ≤ f z + K * dist x z := ciInf_le (mcShane_bddBelow hs hf x) _ |
| _ ≤ f z + K * dist y z + K * dist x y := by |
| rw [add_assoc, ← mul_add, add_comm (dist y z)] |
| gcongr |
| apply dist_triangle |
|
|
| lemma mcShane_dist_le {s : Set E} (hs : s.Nonempty) {K : ℝ≥0} {f g : E → ℝ} |
| (hf : LipschitzOnWith K f s) (hg : LipschitzOnWith K g s) |
| {ε : ℝ} (hε : ∀ x ∈ s, dist (f x) (g x) ≤ ε) (x : E) : |
| dist (mcShane s K f x) (mcShane s K g x) ≤ ε := by |
| let : Nonempty s := hs.to_subtype |
| have aux {f g : E → ℝ} (hf : LipschitzOnWith K f s) |
| (h : ∀ z ∈ s, f z ≤ g z + ε) : |
| mcShane s K f x - ε ≤ mcShane s K g x := by |
| apply le_ciInf |
| intro z |
| have H := ciInf_le (mcShane_bddBelow hs hf x) z |
| change mcShane s K f x ≤ f z + K * dist x z at H |
| linarith [h z z.2] |
| rw [Real.dist_eq, abs_le] |
| have H₁ := aux (g := g) hf (fun z hz => by have H := (abs_le.mp (hε z hz)).2; linarith) |
| have H₂ := aux (g := f) hg (fun z hz => by have H := (abs_le.mp (hε z hz)).1; linarith) |
| constructor <;> linarith |
|
|
| lemma exists_convergent_lipschitz_extensions {s : Set E} (hs : TotallyBounded s) |
| {K : ℝ≥0} {fs : ℕ → E → ℝ} {f : E → ℝ} |
| (hfs : ∀ j, LipschitzOnWith K (fs j) s) |
| (hl : ∀ x ∈ s, Tendsto (fun j => fs j x) atTop (𝓝 (f x))) : |
| ∃ (Fs : ℕ → E → ℝ) (F : E → ℝ), |
| (∀ j, LipschitzWith K (Fs j)) ∧ LipschitzWith K F ∧ |
| (∀ j, EqOn (fs j) (Fs j) s) ∧ EqOn f F s ∧ |
| ∀ x, Tendsto (fun j => Fs j x) atTop (𝓝 (F x)) := by |
| rcases s.eq_empty_or_nonempty with rfl | hsne |
| · exact ⟨fun _ _ => 0,fun _ => 0,fun _ => (LipschitzWith.const _).weaken (by positivity), |
| (LipschitzWith.const _).weaken (by positivity),fun _ => eqOn_empty _ _, |
| eqOn_empty _ _,fun _ => tendsto_const_nhds⟩ |
| have hf := lipschitzOn_limit hfs hl |
| refine ⟨fun j => mcShane s K (fs j),mcShane s K f, |
| fun j => mcShane_lipschitz hsne (hfs j),mcShane_lipschitz hsne hf, |
| fun j => mcShane_eqOn hsne (hfs j),mcShane_eqOn hsne hf,?_⟩ |
| intro x |
| apply Metric.tendsto_atTop.mpr |
| intro ε hε |
| have HU := Metric.tendstoUniformlyOn_iff.mp (uniform_limit_on_totallyBounded hs hfs hl) |
| (ε/2) (half_pos hε) |
| obtain ⟨N,hN⟩ := eventually_atTop.mp HU |
| refine ⟨N,fun j hj => ?_⟩ |
| have HH := mcShane_dist_le hsne hf (hfs j) (fun z hz => (hN j hj z hz).le) x |
| rw [dist_comm] at HH |
| exact HH.trans_lt (half_lt_self hε) |
|
|
| end SharpIntegralFillings |
|
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| end |
|
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| end OAI |
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