On isoperimetric sets of radially symmetric measures

TL;DR

This paper investigates the shape of optimal regions for minimizing perimeter under radially symmetric measures, using symmetrization, variational methods, and numerical solutions to characterize isoperimetric sets.

Contribution

It introduces a reduction of the isoperimetric problem to a second-order ODE and provides empirical and theoretical results for radially symmetric exponential power laws and log-convex measures.

Findings

Large symmetric balls are isoperimetric for strictly log-convex measures.

Numerical solutions describe isoperimetric regions for exponential power laws.

Isoperimetric inequalities are established for log-convex measures.

Abstract

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.