Bound the volume by the mean curvature of the boundary

Question

Let $M$ be a compact compact Riemannian manifold with non-negative Ricci curvature and smooth boundary $N$. Assume that the mean curvature $H_N$ of $N$ is positive.

Question: How to determine a constant $C$ such that $$\int_N \frac{1}{H_N}\geq C\cdot \mathrm{Vol}(M)$$ and the equality holds if and only if $M$ is a ball in an Euclidean space?

It seems like $C=\frac{\dim M+1}{\dim M}$, however, I failed to prove such an inequality in this case.

Answer 1

Let $N_r$ be the level set $\mathrm{dist}_N=r$, and $\pi_r\colon N_r\to N$ is the normal projection; it is uniquely defined almost everywhere. Notice that the Jacobian of $\pi_r$ at $x$ can be estimated from $r$ and $H(\pi_r(x))$ with the equality case for the round ball. Integrating you get a bound on area of $N_r$. It remains to apply the coarea formula.