Hausdorff dimension of the set of endpoints of typical convex surfaces

Alain Rivière (1)

1 LAMFA - Laboratoire Amiénois de Mathématique Fondamentale et Appliquée

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.

Mots clés: cut locus; Hausdorff dimension; convex body

Domaines: Géométrie métrique [math.MG]

Alain Rivière. Hausdorff dimension of the set of endpoints of typical convex surfaces. 2013. ⟨hal-00914708⟩