ScaledRadialEnvironment.lean`

GitHub source file

1 declarations, 0 contain sorry, 0 declared as axioms

A common regular neighborhood with uniform radial estimates

theorem OAI.RegularPointCoordinates.exists_scaled_radial_environment line 19
Source documentation
The published inputs yield one actual heat and Poisson environment, with the radial constants fixed before all requested smallness tolerances.
```
theorem exists_scaled_radial_environment {K : ℝ} (m : ℕ)
    (hX : NoncollapsedRCD K (m + 2) X) [(hausdorff (m + 2) X).Regular]
    {p : X} {a κ : ℝ} (ha : 0 ≤ a) (hκ : 0 ≤ κ) (hK : -κ ≤ K)
    (hcompact : PointedRescalingPrecompactness p) (hregular : IsRegular (m + 2) p)
    (hvolume : EuclideanRescalingVolumeConvergence (hausdorff (m + 2) X) (m + 2) p
      (euclideanUnitVolume (m + 2)))
    (hmeasured : EuclideanMeasuredRescalingConvergence (hausdorff (m + 2) X) (m + 2) p)
    (hBG : BallVolumeComparison (hausdorff (m + 2) X) (comparisonModelVolume (m + 1) a))
    (hdensity : ∀ y, SmallScaleVolumeUpper (hausdorff (m + 2) X)
      (comparisonModelVolume (m + 1) a) y (euclideanUnitVolume (m + 2)))
    (hdistance : LocalSquaredDistanceInput hX) (hsobolev : LocalDirichletSobolevInput hX)
    (hpoisson : LocalPoissonGradientInput hX)
…
```
Mentions: OAI.RegularPointCoordinates.NoncollapsedRCD, OAI.RegularPointCoordinates.hausdorff, OAI.RegularPointCoordinates.PointedRescalingPrecompactness, OAI.RegularPointCoordinates.IsRegular, OAI.RegularPointCoordinates.EuclideanRescalingVolumeConvergence, OAI.RegularPointCoordinates.euclideanUnitVolume, OAI.RegularPointCoordinates.EuclideanMeasuredRescalingConvergence, OAI.RegularPointCoordinates.BallVolumeComparison, OAI.RegularPointCoordinates.comparisonModelVolume, OAI.RegularPointCoordinates.SmallScaleVolumeUpper, OAI.RegularPointCoordinates.LocalSquaredDistanceInput, OAI.RegularPointCoordinates.LocalDirichletSobolevInput, OAI.RegularPointCoordinates.LocalPoissonGradientInput