Hausdorff Dimension of the Set of Endpoints of Typical Convex Surfaces

We mainly prove that most d-dimensional convex surfaces Σ have a set of endpoints of Hausdorff dimension at least d/3. An endpoint means a point not lying in the interior of any shorter path in Σ. "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 a previous paper of the author [Hausdorff dimension of cut loci of generic subspaces of Euclidean spaces, J. Convex Analysis 14 (2007) 823-854] about a similar question. However, our result here is just an estimation about a still unsolved question, as much as we know.