Abstract

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.

Author information

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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

Alain.Riviere@u-picardie.fr

Keywords

  • Cut locus
  • Hausdorff dimension
  • convex body

Mathematics Subject Classification

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

References

  1. K. Adiprasito: Infinite curvature on typical convex surfaces, Geom. Dedicata 159 (2012) 267–275.https://doi.org/10.1007/s10711-011-9658-0Google Scholar
  2. K. Adiprasito, T. Zamfirescu: Large curvature on typical convex surfaces, J. Convex Analysis 19(2) (2012) 385–391.https://doi.org/10.68381/jca19021Google Scholar
  3. A. D. Alexsandrov: Almost everywhere existence of the second differential of a convex function and some properties of convex surfaces connected with it, Leningr. Gos. Univ., Uch. Zap., Ser. Mat. Nauk 6 (1939) 3–35 (in Russian).Google Scholar
  4. A. D. Alexandrov: A. D. Alexandrov Selected Works. Part II: Intrinsic Geometry of Convex Surfaces, S. S. Kutateladze (ed.), Chapman & Hall/CRC, Boca Raton (2006).Google Scholar
  5. K. Falconer: Fractal Geometry, John Wiley & Sons, Chichester (1990).Google Scholar
  6. P. M. Gruber: Baire categories in convexity, in: P. M. Gruber, J. Wills (eds.), Handbook of Convex Geometry, Volume B, North-Holland, Amsterdam (1993) 1327–1346.https://doi.org/10.1016/B978-0-444-89597-4.50021-4Google Scholar
  7. V. Klee: Some new results on smoothness and rotundity in normed linear spaces, Math. Ann. 139 (1959) 51–63.https://doi.org/10.1007/BF01459822Google Scholar
  8. Y. Otsu, T. Shioya: The Riemannian structure of Alexandrov spaces, J. Differ. Geom. 39(3) (1994) 629–658.https://doi.org/10.4310/jdg/1214455075Google Scholar
  9. A. Rivière: Hausdorff dimension of cut loci of generic subspaces of Euclidean spaces, J. Convex Analysis 14(4) (2007) 823–854.https://doi.org/10.68381/jca14047Google Scholar
  10. J. Rouyer: Generic properties of compact metric spaces, Topology Appl. 158(16) (2011) 2140–2147.https://doi.org/10.1016/j.topol.2011.07.003Google Scholar
  11. J.-A. Wieacker: The convex hull of a typical compact set, Math. Ann. 282 (1988) 637–644.https://doi.org/10.1007/BF01462889Google Scholar
  12. T. Zamfirescu: The curvature of most convex surfaces vanishes almost everywhere, Math. Z. 174 (1980) 135–139.https://doi.org/10.1007/BF01293533Google Scholar
  13. T. Zamfirescu: Nonexistence of curvature in most points of most convex surfaces, Math. Ann. 252 (1980) 217–219.https://doi.org/10.1007/BF01420084Google Scholar
  14. T. Zamfirescu: Many endpoints and few interior points of geodesics, Invent. Math. 69(1) (1982) 253–257.https://doi.org/10.1007/BF01399505Google Scholar
  15. T. Zamfirescu: The strange aspect of most compacta, J. Math. Soc. Japan 57(3) (2005) 701–708.https://doi.org/10.2969/jmsj/1158241930Google Scholar

Suggested citation

A. Rivière. “Hausdorff Dimension of the Set of Endpoints of Typical Convex Surfaces.” Journal of Convex Analysis 22 (2015), No. 2, 541–551. https://doi.org/10.68381/jca22026

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Published by Heldermann Verlag, 2015. Rights now held by Banach Press LLC.