| import OAI.Geometry.SingularHomology.SimplexSubdivisionMesh | |
| namespace OAI | |
| noncomputable section | |
| open Set Metric | |
| namespace Problem340.SimplexSubdivision | |
| variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] | |
| def BarycentricCell : (n : ℕ) → (Fin (n + 1) → E) → (Fin (n + 1) → E) → Prop | |
| | 0, v, w => w = v | |
| | n + 1, v, w => ∃ i : Fin (n + 2), ∃ u : Fin (n + 1) → E, | |
| BarycentricCell n (v ∘ i.succAbove) u ∧ | |
| w = Fin.cons (faceCenter v Finset.univ) u | |
| lemma meshRatio_monotone_step (n : ℕ) : | |
| (n : ℝ) / (n + 1) ≤ (n + 1) / (n + 2) := by | |
| apply (div_le_div_iff₀ (by positivity) (by positivity)).2 | |
| nlinarith | |
| theorem BarycentricCell.carrier_and_mesh {n : ℕ} {v w : Fin (n + 1) → E} | |
| (h : BarycentricCell n v w) (D : ℝ) (hD0 : 0 ≤ D) | |
| (hD : ∀ i j, dist (v i) (v j) ≤ D) : | |
| Set.range w ⊆ convexHull ℝ (Set.range v) ∧ | |
| ∀ i j, dist (w i) (w j) ≤ ((n : ℝ) / (n + 1)) * D := by | |
| induction n with | |
| | zero => | |
| change w = v at h | |
| subst w | |
| refine ⟨subset_convexHull ℝ _, ?_⟩ | |
| intro i j | |
| have hij : i = j := Fin.ext (by omega) | |
| simp [hij] | |
| | succ n ih => | |
| obtain ⟨i, u, hu, rfl⟩ := h | |
| obtain ⟨hcarrier, hmesh⟩ := ih hu (fun j k => hD _ _) | |
| have hface : Set.range (v ∘ i.succAbove) ⊆ Set.range v := by | |
| rintro _ ⟨j, rfl⟩ | |
| exact ⟨i.succAbove j, rfl⟩ | |
| have hchild : Set.range u ⊆ convexHull ℝ (Set.range v) := | |
| hcarrier.trans (convexHull_mono hface) | |
| have hcenter : faceCenter v Finset.univ ∈ convexHull ℝ (Set.range v) := by | |
| simpa only [Finset.coe_univ, Set.image_univ] using | |
| faceCenter_mem_convexHull v (Finset.univ_nonempty : (Finset.univ : Finset (Fin (n + 2))).Nonempty) | |
| have hrad (j : Fin (n + 1)) : | |
| dist (faceCenter v Finset.univ) (u j) ≤ | |
| (((n + 1 : ℕ) : ℝ) / ((n + 1 : ℕ) + 1)) * D := by | |
| apply dist_faceCenter_of_mem_convexHull_le_mesh v Finset.univ_nonempty | |
| (n + 1) (by simp) D hD0 (fun a _ b _ => hD a b) | |
| simpa only [Finset.coe_univ, Set.image_univ] using hchild (Set.mem_range_self j) | |
| refine ⟨?_, ?_⟩ | |
| · rintro _ ⟨j, rfl⟩ | |
| refine Fin.cases ?_ (fun k => ?_) j | |
| · exact hcenter | |
| · exact hchild (Set.mem_range_self k) | |
| · intro j k | |
| refine Fin.cases ?_ (fun j => ?_) j | |
| · refine Fin.cases ?_ (fun k => ?_) k | |
| · simp only [Fin.cons_zero, dist_self] | |
| positivity | |
| · simpa only [Fin.cons_zero, Fin.cons_succ] using hrad k | |
| · refine Fin.cases ?_ (fun k => ?_) k | |
| · simpa only [Fin.cons_zero, Fin.cons_succ, dist_comm] using hrad j | |
| · simp only [Fin.cons_succ] | |
| apply (hmesh j k).trans | |
| apply mul_le_mul_of_nonneg_right _ hD0 | |
| simpa only [Nat.cast_add, Nat.cast_one, add_assoc, one_add_one_eq_two] using | |
| meshRatio_monotone_step n | |
| theorem BarycentricCell.carrier {n : ℕ} {v w : Fin (n + 1) → E} | |
| (h : BarycentricCell n v w) : Set.range w ⊆ convexHull ℝ (Set.range v) := by | |
| exact (h.carrier_and_mesh (Metric.diam (Set.range v)) Metric.diam_nonneg | |
| (fun i j => Metric.dist_le_diam_of_mem (Set.finite_range v).isBounded | |
| (Set.mem_range_self i) (Set.mem_range_self j))).1 | |
| theorem BarycentricCell.diam_le {n : ℕ} {v w : Fin (n + 1) → E} | |
| (h : BarycentricCell n v w) : | |
| Metric.diam (convexHull ℝ (Set.range w)) ≤ | |
| ((n : ℝ) / (n + 1)) * Metric.diam (convexHull ℝ (Set.range v)) := by | |
| rw [convexHull_diam, convexHull_diam] | |
| have hmesh := (h.carrier_and_mesh (Metric.diam (Set.range v)) Metric.diam_nonneg | |
| (fun i j => Metric.dist_le_diam_of_mem (Set.finite_range v).isBounded | |
| (Set.mem_range_self i) (Set.mem_range_self j))).2 | |
| apply Metric.diam_le_of_forall_dist_le (by positivity) | |
| rintro _ ⟨i, rfl⟩ _ ⟨j, rfl⟩ | |
| exact hmesh i j | |
| end Problem340.SimplexSubdivision | |
| end | |
| end OAI | |