*preprint*

**Inserted:** 12 nov 2021

**Year:** 2017

**Abstract:**

In this paper the author studies the isoperimetric problem in $\re^n$ with
perimeter density $

x

^p$ and volume density $1.$ We settle completely the case
$n=2,$ completing a previous work by the author: we characterize the case of
equality if $0\leq p\leq 1$ and deal with the case $-\infty<p<-1$ (with the
additional assumption $0\in\Omega$). In the case $n\geq 3$ we deal mainly with
the case $-\infty<p<0,$ showing among others that the results in $2$ dimensions
do not generalize for the range $-n+1<p<0.$