| import Mathlib |
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| namespace OAI |
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| section |
| open Set Filter MeasureTheory |
| open scoped Topology ENNReal NNReal |
| open Filter Set |
| open scoped Topology NNReal |
| open Set Filter MeasureTheory TopologicalSpace |
| open scoped Topology ENNReal |
| open MeasureTheory Filter Set Metric |
| open scoped Topology Pointwise NNReal |
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| namespace SharpIntegralFillings |
| universe u |
| variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] |
| (μ : Measure E) [Measure.IsAddHaarMeasure μ] |
| lemma tangentConeAt_pi_eq_univ_of_density_one {ι : Type*} [Fintype ι] |
| (s : Set E) (x : E) |
| (hx : Tendsto (fun r => μ (s ∩ closedBall x r) / μ (closedBall x r)) |
| (𝓝[>] 0) (𝓝 1)) : |
| tangentConeAt ℝ {f : ι → E | ∀ i, f i ∈ s} (fun _ => x) = univ := by |
| classical |
| apply eq_univ_of_forall |
| intro z |
| rw [tangentConeAt_eq_biInter_closure] |
| simp only [mem_iInter] |
| intro U hU |
| rw [Metric.mem_closure_iff] |
| intro ε hε |
| have B₁ : ∀ᶠ r in 𝓝[>] (0 : ℝ), |
| ∀ i, (s ∩ ({x} + r • closedBall (z i) (ε / 2))).Nonempty := by |
| rw [eventually_all] |
| intro i |
| exact Measure.eventually_nonempty_inter_smul_of_density_one μ s x hx _ |
| measurableSet_closedBall (measure_closedBall_pos μ (z i) (half_pos hε)).ne' |
| have B₂ : ∀ᶠ r in 𝓝[>] (0 : ℝ), r • closedBall z (ε / 2) ⊆ U := by |
| apply nhdsWithin_le_nhds |
| have h := eventually_singleton_add_smul_subset (𝕜 := ℝ) (x := (0 : ι → E)) |
| (s := closedBall z (ε / 2)) isBounded_closedBall hU |
| filter_upwards [h] with r hr |
| simpa using hr |
| obtain ⟨r, hr₁, hrU, hr⟩ := |
| (B₁.and (B₂.and self_mem_nhdsWithin)).exists |
| have H (i : ι) : ∃ a ∈ closedBall (z i) (ε / 2), x + r • a ∈ s := by |
| obtain ⟨y, hys, hy⟩ := hr₁ i |
| simp only [mem_smul_set, image_add_left, mem_preimage, singleton_add] at hy |
| rcases hy with ⟨a, ha, h⟩ |
| refine ⟨a, ha, ?_⟩ |
| simpa only [h, add_neg_cancel_left] using hys |
| choose a ha has using H |
| have haz : a ∈ closedBall z (ε / 2) := by |
| rw [mem_closedBall, dist_pi_le_iff (half_pos hε).le] |
| exact ha |
| refine ⟨a, ?_, ?_⟩ |
| · have hra : r • a ∈ U ∩ ((fun _ : ι => x) + ·) ⁻¹' {f | ∀ i, f i ∈ s} := by |
| exact ⟨hrU (smul_mem_smul_set haz), has⟩ |
| exact ⟨r⁻¹, mem_univ _, r • a, hra, inv_smul_smul₀ (ne_of_gt hr) a⟩ |
| · exact lt_of_le_of_lt (by simpa [dist_comm] using haz) (half_lt_self hε) |
|
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| omit [MeasurableSpace E] [BorelSpace E] in |
|
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| lemma seminorm_continuous_finiteDimensional (p : Seminorm ℝ E) : Continuous p := by |
| classical |
| let b := Module.finBasis ℝ E |
| have hsum (I : Finset (Fin (Module.finrank ℝ E))) |
| (q : Fin (Module.finrank ℝ E) → E) : |
| p (∑ i ∈ I, q i) ≤ ∑ i ∈ I, p (q i) := by |
| induction I using Finset.induction_on with |
| | empty => simp |
| | @insert i I hi ih => |
| simp only [Finset.sum_insert hi] |
| exact (map_add_le_add p _ _).trans (add_le_add le_rfl ih) |
| have hb (v : E) : p v ≤ ∑ i, ‖b.repr v i‖ * p (b i) := by |
| convert hsum Finset.univ (fun i => b.repr v i • b i) using 1 |
| · rw [b.sum_repr] |
| · simp only [map_smul_eq_mul] |
| have hcont : Continuous (fun v => ∑ i, ‖b.repr v i‖ * p (b i)) := by |
| apply continuous_finsetSum |
| intro i _ |
| exact ((continuous_apply i).comp b.continuous_coe_repr).norm.mul_const _ |
| apply Seminorm.continuous_of_continuousAt_zero |
| have hc : Tendsto (fun v => ∑ i, ‖b.repr v i‖ * p (b i)) (𝓝 0) (𝓝 0) := by |
| simpa using hcont.tendsto 0 |
| simpa [ContinuousAt] using tendsto_of_tendsto_of_tendsto_of_le_of_le |
| tendsto_const_nhds hc (fun v => apply_nonneg p v) hb |
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| end SharpIntegralFillings |
| end |
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| end OAI |
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