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Journal of Convex Analysis 22 (2015), No. 2, 541--551
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 Saint-Leu, 80 039 Amiens Cedex 1, France

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.

Keywords: Cut locus, Hausdorff dimension, convex body.

MSC: 28A78, 28A80, 53C22, 54E52, 52A20

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