On isoperimetric constants for log-concave probability distributions

Here, A is an arbitrary Borel subset of R of measure μ(A) with μ-perimeter μ(A) = lime↓0 μ(Ae)−μ(A) e , where Ae = {x ∈ R : |x − a| < e, for some a ∈ A} denotes an open e-neighbourhood of A with respect to the Euclidean distance. The quantity h(μ) represents an important geometric characteristic of the measure and is deeply related to a number of interesting analytic inequalities. As an example, one may consider a Poincare-type inequality ∫ |∇f | dμ ≥ λ1 ∫ |f | dμ

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