I want to ask if the isoperimetric inequality of the following form holds.
Let's suppose that $(\mathbb{R}^n, g)$ is a Riemannian manifold. Let $A \subset \mathbb{R}^n$ with $0 < Vol_g(A) < \infty$ and let $B_g(0, r)$ be the ball (with respect to $g$) centered at the origin with the radius $r$ such that $Vol_g(A) = Vol_g(B_g(0, r))$. Then $Per_g(A) \geq Per_g(B_g(0,r))$.
I think this is true but I couldn't find any literature. Please give any advice. Thanks in advance.