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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 | 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
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