| import OAI.Geometry.SingularHomology.SubdivisionSupportMesh |
| import OAI.Geometry.SingularHomology.BarycentricSubdivision |
| import OAI.Geometry.SingularHomology.SubdivisionSubtypeMesh |
| import OAI.Geometry.SingularHomology.SimplexSubtypeMesh |
|
|
| namespace OAI |
|
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| noncomputable section |
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|
| open Set |
|
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| namespace Problem340.SingularSubdivision |
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| lemma mem_support_linearMap {α β : Type*} |
| (L : (α →₀ ℝ) →ₗ[ℝ] (β →₀ ℝ)) (c : α →₀ ℝ) {q : β} |
| (h : q ∈ (L c).support) : |
| ∃ p ∈ c.support, q ∈ (L (Finsupp.single p 1)).support := by |
| classical |
| have hL : L = Finsupp.linearCombination ℝ (fun p => L (Finsupp.single p 1)) := by |
| apply Finsupp.lhom_ext |
| intro p r |
| simp only [Finsupp.linearCombination_single] |
| have hr : Finsupp.single p r = r • Finsupp.single p 1 := by simp |
| rw [hr, map_smul] |
| rw [hL, Finsupp.linearCombination_apply, Finsupp.sum] at h |
| obtain ⟨p, hp, hq⟩ := Finsupp.mem_support_finsetSum q h |
| exact ⟨p, hp, Finsupp.support_smul hq⟩ |
|
|
| variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
|
|
| theorem iterate_subdivision_support_carrier_and_mesh {n : ℕ} |
| (v : Fin (n + 1) → E) (D : ℝ) (hD : 0 ≤ D) |
| (hvertices : ∀ i j, dist (v i) (v j) ≤ D) |
| (k : ℕ) {w : Fin (n + 1) → E} |
| (h : w ∈ ((subdivision barycentricCenters (n + 1))^[k] (generator v)).support) : |
| range w ⊆ convexHull ℝ (range v) ∧ |
| ∀ i j, dist (w i) (w j) ≤ ((n : ℝ) / (n + 1)) ^ k * D := by |
| induction k generalizing w with |
| | zero => |
| have hw : w = v := by simpa [generator] using h |
| subst w |
| exact ⟨subset_convexHull ℝ _, by simpa using hvertices⟩ |
| | succ k ih => |
| rw [Function.iterate_succ_apply'] at h |
| obtain ⟨u, hu, hw⟩ := mem_support_linearMap _ _ h |
| obtain ⟨huc, hum⟩ := ih hu |
| obtain ⟨hwc, hwm⟩ := (subdivision_support_barycentricCell u hw).carrier_and_mesh |
| (((n : ℝ) / (n + 1)) ^ k * D) (by positivity) hum |
| refine ⟨hwc.trans ?_, ?_⟩ |
| · exact convexHull_min huc (convex_convexHull ℝ _) |
| · intro i j |
| calc |
| dist (w i) (w j) ≤ (n : ℝ) / (n + 1) * |
| (((n : ℝ) / (n + 1)) ^ k * D) := hwm i j |
| _ = ((n : ℝ) / (n + 1)) ^ (k + 1) * D := by ring |
|
|
| theorem iterate_subdivision_support_mesh {n : ℕ} |
| (v : Fin (n + 1) → E) (D : ℝ) (hD : 0 ≤ D) |
| (hvertices : ∀ i j, dist (v i) (v j) ≤ D) |
| (k : ℕ) {w : Fin (n + 1) → E} |
| (h : w ∈ ((subdivision barycentricCenters (n + 1))^[k] (generator v)).support) |
| (i j : Fin (n + 1)) : |
| dist (w i) (w j) ≤ ((n : ℝ) / (n + 1)) ^ k * D := |
| (iterate_subdivision_support_carrier_and_mesh v D hD hvertices k h).2 i j |
|
|
| theorem iterate_subdivision_support_carrier {n : ℕ} |
| (v : Fin (n + 1) → E) (k : ℕ) {w : Fin (n + 1) → E} |
| (h : w ∈ ((subdivision barycentricCenters (n + 1))^[k] (generator v)).support) : |
| range w ⊆ convexHull ℝ (range v) := |
| (iterate_subdivision_support_carrier_and_mesh v (Metric.diam (range v)) |
| Metric.diam_nonneg (fun i j => Metric.dist_le_diam_of_mem |
| (finite_range v).isBounded (mem_range_self i) (mem_range_self j)) k h).1 |
|
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| theorem simplex_iterate_subdivision_support_mesh {m n : ℕ} |
| (v : Fin (n + 1) → StandardSimplex m) (D : ℝ) (hD : 0 ≤ D) |
| (hvertices : ∀ i j, dist (v i) (v j) ≤ D) |
| (k : ℕ) {w : Fin (n + 1) → StandardSimplex m} |
| (h : w ∈ ((subdivision (simplexCenters m) (n + 1))^[k] (generator v)).support) |
| (i j : Fin (n + 1)) : |
| dist (w i) (w j) ≤ ((n : ℝ) / (n + 1)) ^ k * D := |
| iterate_subdivision_support_mesh (fun a => RealSimplex.coordinates (v a)) D hD hvertices k |
| (simplex_iterated_subdivision_support v h) i j |
|
|
| theorem simplex_iterate_subdivision_realize_dist {m n : ℕ} |
| (v : Fin (n + 1) → StandardSimplex m) (D : ℝ) (hD : 0 ≤ D) |
| (hvertices : ∀ i j, dist (v i) (v j) ≤ D) |
| (k : ℕ) {w : Fin (n + 1) → StandardSimplex m} |
| (h : w ∈ ((subdivision (simplexCenters m) (n + 1))^[k] (generator v)).support) |
| (x y : StandardSimplex n) : |
| dist (AffineSimplex.simplexRealizeMap w x) (AffineSimplex.simplexRealizeMap w y) ≤ |
| ((n : ℝ) / (n + 1)) ^ k * D := by |
| apply SimplexSubtypeMesh.dist_le_of_convex_carrier (fun i => RealSimplex.coordinates (w i)) |
| (fun point => RealSimplex.coordinateHomeomorph (AffineSimplex.simplexRealizeMap w point)) |
| _ _ _ x y |
| · exact simplex_iterate_subdivision_support_mesh v D hD hvertices k h |
| · intro a |
| simpa only [RealSimplex.coordinateHomeomorph_apply, RealSimplex.coordinates, |
| AffineSimplex.simplexRealizeMap_coe] using |
| AffineSimplex.realize_mem_convexHull (fun i => RealSimplex.coordinates (w i)) a |
|
|
| theorem standardVertices_dist_le_one (n : ℕ) (i j : Fin (n + 1)) : |
| dist (standardVertices n i) (standardVertices n j) ≤ 1 := by |
| exact (Metric.dist_le_diam_of_mem |
| (Convexity.StdSimplex.isBounded_range_toFun_comp_weights (Fin (n + 1))) |
| (mem_range_self (standardVertices n i)) (mem_range_self (standardVertices n j))).trans |
| (Convexity.StdSimplex.diam_range_toFun_comp_weights_subset_closedBall (Fin (n + 1))) |
|
|
| open Filter |
| open scoped Topology |
|
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| theorem exists_iterated_subdivision_small_images |
| {X I : Type} [TopologicalSpace X] (n : ℕ) |
| (S : Finset C(StandardSimplex n, X)) (U : I → Set X) |
| (hopen : ∀ i, IsOpen (U i)) (hcover : (univ : Set X) ⊆ ⋃ i, U i) : |
| ∃ N : ℕ, ∀ k ≥ N, ∀ σ ∈ S, |
| ∀ w ∈ ((subdivision (simplexCenters n) (n + 1))^[k] |
| (generator (standardVertices n))).support, |
| ∃ i, range (σ.comp (AffineSimplex.simplexRealizeMap w)) ⊆ U i := by |
| obtain ⟨δ, hδ, hsmall⟩ := SingularSimplexLebesgue.exists_finite_reparametrization_radius |
| S U hopen (fun _ _ => subset_trans (subset_univ _) hcover) |
| have hr0 : (0 : ℝ) ≤ (n : ℝ) / (n + 1) := by positivity |
| have hr1 : (n : ℝ) / (n + 1) < 1 := by |
| apply (div_lt_one (by positivity)).2 |
| linarith |
| have hlim := tendsto_pow_atTop_nhds_zero_of_lt_one hr0 hr1 |
| obtain ⟨N, hN⟩ := eventually_atTop.mp (hlim.eventually (gt_mem_nhds hδ)) |
| refine ⟨N, ?_⟩ |
| intro k hk σ hσ w hw |
| apply hsmall σ hσ (AffineSimplex.simplexRealizeMap w) |
| intro x y |
| have hdist := simplex_iterate_subdivision_realize_dist (standardVertices n) 1 |
| zero_le_one (standardVertices_dist_le_one n) k hw x y |
| exact hdist.trans_lt (by simpa only [mul_one] using hN k hk) |
|
|
| end Problem340.SingularSubdivision |
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| end |
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| end OAI |
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