We mainly prove that most $d$-dimensional convex surfaces $\Sigma$ have a set of endpoints of Hausdorff dimension at least $d/3$. An \emph{endpoint} means a point not lying in the interior of any shorter path in $\Sigma$. ''Most'' means that the exceptions constitute a meager set, relatively to the usual Hausdorff-Pompeiu distance. The proof employs some of the ideas used in \cite{Riviere07JCA} about a similar question. However, our result here is just an estimation about a still unsolved question, as much as we know.
