Borell and Landau--Shepp inequalities for
Cauchy-type~measures
Tomasz Byczkowski
Tomasz Żak
Abstract:
We investigate various inequalities for the one-dimensional Cauchy measure. We also consider analogous
properties for one-dimensional sections of multidimensional isotropic Cauchy measures.
The paper is a continuation of our previous investigations ,
where we found, among intervals with fixed measure, the ones with the extremal measure of the boundary.
Here for the above mentioned measures we investigate inequalities that are analogous to those proved
for Gaussian measures by Borell and by Landau and
Shepp . We also consider a 1-symmetrization for
Cauchy measures, analogous to the one defined for Gaussian measures by Ehrhard ,
and we prove the concavity of this operation.
2010 AMS Mathematics Subject Classification: Primary 60E05, 60E07. Secondary
Keywords and phrases: auchy distribution, Borell inequality, Landau--Shepp inequality,
Ehrhard symmetrization.
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