  We study the isoperimetric problem for the radially symmetric measures.
Applying the spherical symmetrization procedure and variational arguments we
reduce this problem to a one-dimensional ODE of the second order. Solving
numerically this ODE we get an empirical description of isoperimetric regions
of the planar radially symmetric exponential power laws. We also prove some
isoperimetric inequalities for the log-convex measures. We show, in particular,
that the symmetric balls of large size are isoperimetric sets for strictly
log-convex and radially symmetric measures. In addition, we establish some
comparison results for general log-convex measures.

Probability;Optimization and Control