Geodesic
Hausdorff Dimension of the Set of Endpoints of Typical Convex Surfaces
Journal of convex analysis, Tome 22 (2015) no. 2, pp. 541-551
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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.
Classification : 28A78, 28A80, 53C22, 54E52, 52A20
Mots-clés : Cut locus, Hausdorff dimension, convex body 
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     author = {A. Rivi\`ere},
     title = {Hausdorff {Dimension} of the {Set} of {Endpoints} of {Typical} {Convex} {Surfaces}},
     journal = {Journal of convex analysis},
     pages = {541--551},
     year = {2015},
     volume = {22},
     number = {2},
     url = {http://geodesic.mathdoc.fr/item/JCA_2015_22_2_JCA_2015_22_2_a11/}
}
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A. Rivière. Hausdorff Dimension of the Set of Endpoints of Typical Convex Surfaces. Journal of convex analysis, Tome 22 (2015) no. 2, pp. 541-551. http://geodesic.mathdoc.fr/item/JCA_2015_22_2_JCA_2015_22_2_a11/