Abstract

Let FF be a closed set of the Euclidean space Ed\Bbb{E}^d, with ∅≠F≠Ed\emptyset\not=F\not=\Bbb{E}^d and d≥2d\geq 2. Let N\mathcal{N} be the set of centers of all open balls contained in Ed∖F\Bbb{E}^d \setminus F which are maximal with respect to inclusion. We prove that the Hausdorff dimension dimH(N)\mathrm{dim_H}(\mathcal{N}) of N\mathcal{N} equals dd when FF is, in the sense of Baire categories, a generic compact subset of Ed\mathbb{E}^d, or when Ed∖F\Bbb{E}^d \setminus F is the interior of a generic convex body of Ed\mathbb{E}^d. If CC is a generic convex body, we deduce that the set of all points of ∂C\partial C where the ``upper curvature'' of ∂C\partial C is positive and finite, is of Hausdorff dimension d−1d-1. Let CurvCt\mathrm{CurvCt} be the set of centers of upper curvature of ∂C\partial C, and ω\omega be any non empty open subset of Ed\Bbb{E}^d. We also prove that dimH(ω∩CurvCt)=d\dimH(\omega\cap \mathrm{CurvCt})=d. Let BB be a generic compact subset of Ed\mathbb{E}^d, or a generic convex body of Ed\mathbb{E}^d. Let aN\aN be the set of centers of all closed balls containing BB which are minimal with respect to inclusion. We also prove that dimH(aN)=d\mathrm{dim_H}(\aN)=d. The proofs employ some of the ideas used in a previous paper of the author [``Dimension de Hausdorff de la nervure'', Geom. Dedicata, 85 (2001) 217–235] to construct large cut loci in Ed\mathbb{E}^d.

Author information

Contact details are reproduced from the original publication and may be historical.

Alain Rivière

Lab. Amiénois de Mathématiques Fondamentales et Appliquées, CNRS, UMR 6140, Faculté de Mathématiques et Informatique d'Amiens, 33 rue Saint-Leu, 80 039 Amiens, France

alain.riviere@u-picardie.fr

Keywords

  • Cut locus
  • skeleton
  • medial axis
  • Hausdorff dimension
  • critical value
  • curvature
  • convex body
  • farthest distance

Mathematics Subject Classification

28A78, 28A80

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Suggested citation

A. Rivière. “Hausdorff Dimension of Cut Loci of Generic Subspaces of Euclidean Spaces.” Journal of Convex Analysis 14 (2007), No. 4, 823–854. https://doi.org/10.68381/jca14047

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Published by Heldermann Verlag, 2007. Rights now held by Banach Press LLC.