| import Mathlib |
| import OAI.Geometry.BallPacking.Continuation.ProperPaths |
| |
| namespace OAI |
|
|
| noncomputable section |
| open scoped ContDiff Topology |
| open Set Function Filter |
| namespace HigherDimensionalBallPacking |
| open Rigidity Rigidity.HolderCompletion |
|
|
| theorem adapted_affine_line {n : ℕ} {J : Phase n → End n} |
| (hJs : ContDiff ℝ ∞ J) (hJ : ∀ x,Compatible (J x)) |
| (hJc : HasCompactSupport (fun x => J x-standardJ n)) |
| (hinside : tsupport (fun x => J x-standardJ n)⊆openBall n 1) |
| (p q : Phase n) (hpq : p≠q) : ∃ u,AffineLineCurve J p q u := by |
| let : NormedAddCommGroup (COne ℂ (Phase n)) := inferInstance |
| let : NormedSpace ℝ (COne ℂ (Phase n)) := inferInstance |
| let : NormedAddCommGroup (HMap ℂ (Phase n)) := inferInstance |
| let : NormedSpace ℝ (HMap ℂ (Phase n)) := inferInstance |
| obtain ⟨β,β₁,M,δ,hpath,hend⟩ := adapted_line_continuation_data hJs hJ hJc hinside p q hpq |
| let : NormedAddCommGroup (markedModel (E := Phase n) (Metric.closedBall (0:ℂ) β.rOut)) := inferInstance |
| let : NormedSpace ℝ (markedModel (E := Phase n) (Metric.closedBall (0:ℂ) β.rOut)) := inferInstance |
| let : NormedAddCommGroup (supportedHolder (E := Phase n) (Metric.closedBall (0:ℂ) β.rOut)) := inferInstance |
| let : NormedSpace ℝ (supportedHolder (E := Phase n) (Metric.closedBall (0:ℂ) β.rOut)) := inferInstance |
| obtain ⟨w,hw,hz⟩ := hpath.endpoint_exists |
| exact ⟨_,hend w hw hz⟩ |
|
|
| theorem normalized_two_ball_bound (n : ℕ) (_ : 3≤n) |
| (a b : ℝ) (ha : 0<a) (hb : 0<b) (hp : HasPacking n 2 1 ![a,b]) : a+b≤1 := by |
| obtain ⟨J,p,q,hJs,hJ,hJc,hinside,hpq,harea⟩ := packing_two_line_reduction ha hb hp |
| obtain ⟨u,hu⟩ := adapted_affine_line hJs hJ hJc hinside p q hpq |
| exact harea u hu |
|
|
| end HigherDimensionalBallPacking |
|
|
| end |
|
|
| end OAI |
|
|
