| import Mathlib |
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| namespace OAI |
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| open MeasureTheory Metric |
| open scoped ENNReal |
| namespace AllScaleInverseVolume |
| local instance namedAtLeastTwoTwo : Nat.AtLeastTwo 2 := ⟨Nat.le_refl 2⟩ |
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|
| abbrev Ambient (d : ℕ) := EuclideanSpace ℝ (Fin d) |
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| 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) |
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| 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) |
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| 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))) |
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| open Set |
| open scoped Topology NNReal |
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| end AllScaleInverseVolume |
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| end OAI |
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