| import OAI.Geometry.RCD.Metric.RescaledBallVolume |
|
|
| namespace OAI |
|
|
| /-! # A positive fixed-radius volume bound on every bounded region -/ |
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|
| open Set Metric MeasureTheory |
| open scoped ENNReal |
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|
| noncomputable section |
| namespace RegularPointCoordinates |
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|
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] |
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|
| theorem NoncollapsedRCD.ball_volume_lower_on_bounded_region {K : ℝ} {n : ℕ} |
| (hX : NoncollapsedRCD K n X) (hn : 2 ≤ n) (p : X) (L r : ℝ) (hr : 0 < r) : |
| ∃ m : ℝ, 0 < m ∧ ∀ y, dist y p ≤ L → m ≤ (hausdorff n X).real (ball y r) := by |
| let B := max r (L+1) |
| have hB : 0 < B := hr.trans_le (le_max_left _ _) |
| have hrB : r/B ≤ 1 := (div_le_one hB).mpr (le_max_left _ _) |
| have hδ : 0 < r/B := div_pos hr hB |
| let a := |K| * B^2 |
| have ha : 0 ≤ a := mul_nonneg (abs_nonneg _) (sq_nonneg _) |
| have hcurv : -a ≤ K*B^2 := by |
| have hh := mul_le_mul_of_nonneg_right (neg_abs_le K) (sq_nonneg B) |
| simpa only [a,neg_mul] using hh |
| let m := ((r/B)/(2*Real.exp (4*Real.sqrt a)))^n*(hausdorff n X).real (ball p B) |
| have hV : 0 < (hausdorff n X).real (ball p B) := |
| ENNReal.toReal_pos (hX.fullSupport p B hB).ne' (hX.finiteOnBounded _ isBounded_ball).ne |
| have hm : 0 < m := mul_pos (pow_pos (div_pos hδ (by positivity)) _) hV |
| refine ⟨m,hm,?_⟩ |
| intro y hy |
| have hyB : dist y p ≤ B := hy.trans (by dsimp only [B]; linarith [le_max_right r (L+1)]) |
| have hh := hX.rescaled_ball_volume_lower hn ha hδ hrB p B hB hcurv y hyB |
| have he : r/B*B = r := div_mul_cancel₀ _ hB.ne' |
| simpa only [he] using hh |
|
|
| end RegularPointCoordinates |
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| end |
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| end OAI |
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|
