Let $(M,g)$ be a complete 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 of same volume than $\Omega$.

Is that true?

What if $\Omega$ is large, how can one define a geodesic ball (say if the radius is larger than $inj(M,g)$).