Mean exit time from convex hypersurfaces

L. Karp and M. Pinsky proved that, for small radius $R$, the mean exit time function $E_{R}$ of an extrinsic $R$-ball in a hypersurface $P^{n-1} \subseteq \mathbb{R}^{n}$ is bounded from below by the corresponding function $\widetilde E_{R}$ defined on an extrinsic $R$-ball in $\mathbb{R}^{n-1}$. A counterexample given by C. Mueller proves that this inequality doesn't holds in the large. In this paper we show that, if $P$ is convex, then the inequality holds for all radii. Moreover, we characterize the equality and show that analogous results are true in the sphere.