A Sharp Comparison Theorem for Compact Manifolds with Mean Convex Boundary

Let $M$ be a compact $n$-dimensional Riemannian manifold with nonnegative Ricci curvature and mean convex boundary $\mathrm{\partial }M$. Assume that the mean curvature $H$ of the boundary $\mathrm{\partial }M$ satisfies $H\ge (n-1)k>0$ for some positive constant $k$. In this paper, we prove that the distance function $d$ to the boundary $\mathrm{\partial }M$ is bounded from above by $\frac{1}{k}$ and the upper bound is achieved if and only if $M$ is isometric to an $n$-dimensional Euclidean ball of radius $\frac{1}{k}$.