File size: 1,595 Bytes
b317718 | 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 | import Mathlib
namespace OAI
open MeasureTheory Metric
open scoped ENNReal
namespace AllScaleInverseVolume
local instance namedAtLeastTwoTwo : Nat.AtLeastTwo 2 := ⟨Nat.le_refl 2⟩
abbrev Ambient (d : ℕ) := EuclideanSpace ℝ (Fin d)
noncomputable local instance ambientSub (d : ℕ) : Sub (Ambient d) where
sub vector other := WithLp.toLp (2 : ℝ≥0∞)
(WithLp.ofLp (p := (2 : ℝ≥0∞)) vector - WithLp.ofLp (p := (2 : ℝ≥0∞)) other)
noncomputable local instance ambientInner (d : ℕ) : Inner ℝ (Ambient d) where
inner vector other := ∑ coordinate : Fin d,
inner ℝ (WithLp.ofLp (p := (2 : ℝ≥0∞)) vector coordinate)
(WithLp.ofLp (p := (2 : ℝ≥0∞)) other coordinate)
noncomputable def gramVolume {d k : ℕ} (u : Ambient d) (z : Fin k → Ambient d) : ℝ :=
Real.sqrt (Matrix.gram ℝ (fun i => z i-u)).det
noncomputable def inverseVolume {d : ℕ} (u : Ambient d) (z : Fin (d/100) → Ambient d) : ℝ≥0∞ :=
((ENNReal.ofReal (gramVolume u z))⁻¹)^(d/100)
def BallBound {d : ℕ} (ν : Measure (Ambient d)) : Prop :=
∀ (x : Ambient d) (r : ℝ), 0 < r →
ν (ball x r) ≤ ENNReal.ofReal (r^(((d:ℝ)-1)/2))
def InverseVolumeStatement : Prop :=
∃ C : ℝ, 0 < C ∧ ∃ d₀ : ℕ, ∀ d : ℕ, d₀ ≤ d →
∀ ν : Measure (Ambient d), IsProbabilityMeasure ν → BallBound ν →
∀ u : Ambient d, (∫⁻ z,inverseVolume u z ∂Measure.pi (fun _ : Fin (d/100) => ν)) ≤
ENNReal.ofReal (Real.exp (C*d*(d/100:ℕ)))
open Set
open scoped Topology NNReal
end AllScaleInverseVolume
end OAI
|