## 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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### References:

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