| import OAI.Geometry.RCD.Transport.ContinuousLengthDistortion |
| import OAI.Geometry.RCD.Sobolev.WeakEndpointWeights |
|
|
| namespace OAI |
|
|
| /-! # Reduced distortion as a bounded continuous path test -/ |
|
|
| open Set Filter MeasureTheory Metric TopologicalSpace |
| open scoped ENNReal NNReal Topology BoundedContinuousFunction |
| noncomputable section |
| namespace RegularPointCoordinates |
| variable {X : Type*} [MetricSpace X] |
|
|
| def pathDistortion (K N t : ℝ) (hK : K≤0) (hN : 1<N) (ht : 0≤t) (ht1 : t≤1) : |
| C(Time,X) →ᵇ ℝ := |
| BoundedContinuousFunction.mkOfBound |
| ⟨fun γ => (sigma K N t (dist (γ 0) (γ 1))).toReal,by |
| have hh : Continuous (fun γ : C(Time,X) => sigma K N t (dist (γ 0) (γ 1))) := |
| (continuous_sigma_nonpos_length hK hN ht ht1).comp |
| ((continuous_eval_const 0).nndist (continuous_eval_const 1)) |
| exact continuous_iff_continuousAt.mpr (fun γ => |
| (ENNReal.continuousAt_toReal (sigma_nonpos_ne_top (t:=t) |
| (d:=dist (γ 0) (γ 1)) hK hN dist_nonneg)).comp |
| (f:=fun η : C(Time,X) => sigma K N t (dist (η 0) (η 1))) hh.continuousAt)⟩ |
| 1 (by |
| intro γ η |
| have hb (ζ : C(Time,X)) : 0≤(sigma K N t (dist (ζ 0) (ζ 1))).toReal ∧ |
| (sigma K N t (dist (ζ 0) (ζ 1))).toReal≤1 := by |
| refine ⟨ENNReal.toReal_nonneg,?_⟩ |
| exact (ENNReal.toReal_mono ENNReal.ofReal_ne_top |
| (sigma_nonpos_le_time hK hN ht ht1 dist_nonneg)).trans |
| (by simpa only [ENNReal.toReal_ofReal ht] using ht1) |
| change |(sigma K N t (dist (γ 0) (γ 1))).toReal- |
| (sigma K N t (dist (η 0) (η 1))).toReal|≤1 |
| exact abs_le.mpr ⟨by linarith [(hb γ).1,(hb η).2],by linarith [(hb γ).2,(hb η).1]⟩) |
|
|
| end RegularPointCoordinates |
|
|
| end |
|
|
| end OAI |
|
|
