The second fundamental form $II$ of $\partial V$ vanishes in the radial direction (along any line that goes through the origin), so one eigenvalue of $II$ is zero.
There are only two eigenvalues since $\dim(\partial V)=2$ and the mean curvature is (up to a factor of 2) the trace of $II$.
Therefore the nonzero eigenvalue of $II$ has the same sign as the mean curvature.

We have thus that $II$ is positive semidefinite iff the mean curvature is positive (and similarly for negative).
Positive semidefiniteness of $II$ is equivalent with convexity, so the answer to your question is yes.