Suppose $\mathcal{M}$ is a Riemannian manifold with metric $g$ and with the volume measure $d\mu$ (induced by $g$). Let $f:\mathcal{M}\to \mathbb{R}$ be $C^{\infty}$. Is it true that $$\max_{\mathcal{M}}f-\min_{\mathcal{M}}f\leq\int_{\mathcal{M}}|\nabla f|d\mu \;?$$ Here $|\nabla f|$ is the norm of the gradient of $f$ with respect to the metric $g.$
Edit 1: Following the comments I'd like to know if the following inequalities hold $$(\max_{\mathcal{M}}f-\min_{\mathcal{M}}f)\operatorname{diam}{({\mathcal{M}})}\leq\int_{\mathcal{M}}|\nabla f|d\mu \;?$$ Edit 2:
$$(\max_{\mathcal{M}}f-\min_{\mathcal{M}}f)\operatorname{Vol}{(\mathcal{M})}\leq\int_{\mathcal{M}}|\nabla f|d\mu \;?$$