Let $(M,g)$ be a complete Riemannian, connected, compact manifold (with or without boundary). Let $f(r)$ be a decreasing function of $r =$ geodesic distance. If $\Omega \subset M$, then $$ \int_{\Omega} f(r)\, dV \leq \int_{\Omega^\star} f(r)\, dV \, ,$$ where $\Omega^\star$ is a geodesic ball with the same volume as $\Omega$.

Is that true? What if $\Omega$ is large? How can one define a geodesic ball, say if the radius is larger than $\mathrm{inj}(M,g)$?