Isoperimetric Regions in $\mathbb{R}^n$ with density $r^p$

We show that the unique isoperimetric hypersurfaces in $\mathbb{R}^n$ with density $r^p$ for $n \ge 3$ and $p>0$ are spheres that pass through the origin.
Comment: published online in Analysis and Geometry in Metric Spaces, revised according to referee report, 29 pages, 10 figures