Some isoperimetric problems in planes with density

TL;DR

This paper investigates isoperimetric problems in the plane with various densities, including piecewise constant and Gaussian, analyzing specific cases and posing open questions for future research.

Contribution

It introduces new results on isoperimetric sets in planes with different densities and explores particular configurations and open problems.

Findings

Characterization of isoperimetric sets for piecewise constant densities

Analysis of isoperimetric problems under Gaussian density

Identification of open questions in the field

Abstract

We study the isoperimetric problem in Euclidean space endowed with a density. We first consider piecewise constant densities and examine particular cases related to the characteristic functions of half-planes, strips and balls. We also consider continuous modification of Gauss density in $\R^2$. Finally, we give a list of related open questions.