EXTREMAL PROPERTIES OF ONE-DIMENSIONAL CAUCHY-TYPE
MEASURES
Abstract: In the paper we investigate some extremal properties of intervals for the
one-dimensional Cauchy measure, related to isoperimetric inequalities. We also consider the
analogous properties for the one-dimensional sections of the multidimensional isotropic
Cauchy measure. In particular, among intervals with the fixed measure we find
the ones with the extremal measure of the boundary. It turns out that, contrary to
the Gaussian case, the type of extremal set depends on the value of the measure.
2000 AMS Mathematics Subject Classification: Primary: 60E05; Secondary:
60E07.
Keywords and phrases: Cauchy distribution, isoperimetry.