
Journal of Convex Analysis 22 (2015), No. 2, 541551 Copyright Heldermann Verlag 2015 Hausdorff Dimension of the Set of Endpoints of Typical Convex Surfaces Alain Rivière Laboratoire Amiénois de Mathématiques Fondamentales et Appliquées, CNRS  UMR 7352, Faculté de Sciences d'Amiens, 33 rue SaintLeu, 80 039 Amiens Cedex 1, France Alain.Riviere@upicardie.fr We mainly prove that most ddimensional 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 HausdorffPompeiu 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) 823854] about a similar question. However, our result here is just an estimation about a still unsolved question, as much as we know. Keywords: Cut locus, Hausdorff dimension, convex body. MSC: 28A78, 28A80, 53C22, 54E52, 52A20 [ Fulltextpdf (154 KB)] for subscribers only. 