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Weighted Poincaré-type inequalities for Cauchy and other convex measures. (English) Zbl 1178.46041

The aim of the present paper is to extend some famous analytic inequalities of Brascamp-Lieb-type, Poincaré-type, and others, to the weighted case. Those inequalities are studied for multi-dimensional Cauchy distributions as well as for more general so-called \(\kappa\)-concave measures. A measure \(\lambda\) is said to be \(\kappa\)-concave for some \(-\infty\leq \kappa\leq \infty\) provided that
\[ \lambda_*(t A +(1-t)B)\geq\left[t\,\lambda(A)^\kappa+(1-t)\lambda(B)^\kappa\right]^{1/\kappa} \]
for \(0<t<1\) and all Borel sets \(A,B\) of positive measure. The case \(\kappa=0\) corresponds to log-concave measures, while \(\kappa=-\infty\) leads to the so-called convex or hyperbolic probability measures. Finally, Cheeger-type isoperimetric inequalities are investigated similarly.

MSC:

46G12 Measures and integration on abstract linear spaces
60B11 Probability theory on linear topological spaces
60G07 General theory of stochastic processes
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