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b37e936 | 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 | import OAI.Geometry.RCD.Metric.RescaledBallVolume
namespace OAI
/-! # Uniform polynomial normalization volumes for centers in one compact ball -/
open Set Filter MeasureTheory Metric
open scoped ENNReal
noncomputable section
namespace RegularPointCoordinates.NoncollapsedRCD
variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X]
{K : ℝ} {n : ℕ} (hX : NoncollapsedRCD K n X) (hn : 2 ≤ n)
include hX hn in
theorem compact_center_volume_power (z : X) {R : ℝ} (hR : 1 ≤ R) :
∃ V : ℝ, 0 < V ∧ ∀ (y : X), dist y z ≤ R →
∀ {t : ℝ}, 0 < t → t ≤ 1 →
V*(Real.sqrt t)^n ≤ (hausdorff n X).real (ball y (Real.sqrt t)) := by
have hR0 : 0 < R := lt_of_lt_of_le (by norm_num) hR
let a := |K| *R^2
have ha : 0 ≤ a := mul_nonneg (abs_nonneg K) (sq_nonneg R)
let E := (2*Real.exp (4*Real.sqrt a))*R
have hE : 0 < E := by dsimp only [E]; positivity
let mass := (hausdorff n X).real (ball z R)
have hmass : 0 < mass := ENNReal.toReal_pos (hX.fullSupport z R hR0).ne'
(hX.finiteOnBounded _ isBounded_ball).ne
let V := mass/E^n
refine ⟨V,div_pos hmass (pow_pos hE n),?_⟩
intro y hy t ht ht1
have hs : 0 < Real.sqrt t := Real.sqrt_pos.mpr ht
have hs1 : Real.sqrt t ≤ R := by
nlinarith [Real.sq_sqrt ht.le,Real.sqrt_nonneg t]
have hcurv : -a ≤ K*R^2 := by
have hh := mul_le_mul_of_nonneg_right (neg_abs_le K) (sq_nonneg R)
simpa only [a,neg_mul] using hh
have hb := hX.rescaled_ball_volume_lower hn ha (div_pos hs hR0)
((div_le_one hR0).mpr hs1) z R hR0 hcurv y hy
have hrad : Real.sqrt t/R*R=Real.sqrt t := by field_simp
have hcoef : (Real.sqrt t/R)/(2*Real.exp (4*Real.sqrt a))=Real.sqrt t/E := by
dsimp only [E]
field_simp
rw [hrad,hcoef] at hb
calc
V*(Real.sqrt t)^n = (Real.sqrt t/E)^n*mass := by
dsimp only [V]
rw [div_pow]
ring
_ ≤ _ := hb
end RegularPointCoordinates.NoncollapsedRCD
end
end OAI
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