| import OAI.Analysis.FreeEntropy.FlowSeparation |
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| namespace OAI |
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| universe u1 |
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| noncomputable section |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal |
| namespace FiniteEntropySeparation |
| theorem gaussian_affine_volume {ι : Type u1} [Fintype ι] (a : EuclideanSpace ℝ ι) {r : ℝ} (hr : 0<r) |
| (s : Set (EuclideanSpace ℝ ι)) |
| (h : (1/2:ℝ≥0∞) ≤ productGaussian ι ((fun x => a+r • WithLp.toLp 2 x) ⁻¹' s)) : |
| ENNReal.ofReal (r^(Fintype.card ι)/2) ≤ volume s := by |
| have hp : productGaussian ι ((fun x => a+r • WithLp.toLp 2 x) ⁻¹' s) ≤ |
| ENNReal.ofReal ((r^(Fintype.card ι))⁻¹)*volume s := by |
| have hh := productGaussian_toLp_le_volume ι ((fun x => a+r • x) ⁻¹' s) |
| have he : ((fun x : EuclideanSpace ℝ ι => a+r • x) ⁻¹' s) = |
| (fun x => r • x) ⁻¹' ((fun x => a+x) ⁻¹' s) := rfl |
| rw [he,Measure.addHaar_preimage_smul volume hr.ne', measure_preimage_add] at hh |
| simpa only [finrank_euclideanSpace,Set.preimage_preimage,Function.comp_def,abs_of_pos (inv_pos.mpr (pow_pos hr _))] using hh |
| have hm : ENNReal.ofReal (r^(Fintype.card ι)) * (1/2:ℝ≥0∞) ≤ |
| ENNReal.ofReal (r^(Fintype.card ι)) * (ENNReal.ofReal ((r^(Fintype.card ι))⁻¹)*volume s) := by |
| gcongr; exact h.trans hp |
| rw [← mul_assoc,← ENNReal.ofReal_mul (pow_nonneg hr.le _),mul_inv_cancel₀ (pow_ne_zero _ hr.ne'),ENNReal.ofReal_one,one_mul] at hm |
| simpa only [div_eq_mul_inv,ENNReal.ofReal_mul (pow_nonneg hr.le _),ENNReal.ofReal_inv_of_pos (by norm_num : (0:ℝ)<2),ENNReal.ofReal_ofNat,one_mul] using hm |
| end FiniteEntropySeparation |
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| end |
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| end OAI |
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