Some isoperimetric problems in planes with density

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 ${\mathbb{R}}^2$. Finally, we give a list of related open questions.

40 pages, 19 figures