Journal of Convex Analysis 14 (2007), No. 4, 823--854
Copyright Heldermann Verlag 2007
Hausdorff Dimension of Cut Loci of Generic Subspaces of Euclidean Spaces
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
[Abstract-pdf]
\def\aN{\mathrm{a}\mathcal{N}} \def\dimH{\mathop{{\rm dim}_{\rm H}}\nolimits} Let $F$ be a closed set of the Euclidean space $\Bbb{E}^d$, with $\emptyset\not=F\not=\Bbb{E}^d$ and $d\geq 2$. Let $\mathcal{N}$ be the set of centers of all open balls contained in $\Bbb{E}^d \setminus F$ which are maximal with respect to inclusion. We prove that the Hausdorff dimension $\mathrm{dim_H}(\mathcal{N})$ of $\mathcal{N}$ equals $d$ when $F$ is, in the sense of Baire categories, a generic compact subset of $\mathbb{E}^d$, or when $\Bbb{E}^d \setminus F$ is the interior of a generic convex body of $\mathbb{E}^d$. If $C$ is a generic convex body, we deduce that the set of all points of $\partial C$ where the ``upper curvature'' of $\partial C$ is positive and finite, is of Hausdorff dimension $d-1$. Let $\mathrm{CurvCt}$ be the set of centers of upper curvature of $\partial C$, and $\omega$ be any non empty open subset of $\Bbb{E}^d$. We also prove that $\dimH(\omega\cap \mathrm{CurvCt})=d$. Let $B$ be a generic compact subset of $\mathbb{E}^d$, or a generic convex body of $\mathbb{E}^d$. Let $\aN$ be the set of centers of all closed balls containing $B$ which are minimal with respect to inclusion. We also prove that $\mathrm{dim_H}(\aN)=d$. 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 $\mathbb{E}^d$.
Keywords: Cut locus, skeleton, medial axis, Hausdorff dimension, critical value, curvature, convex body, farthest distance.
MSC: 28A78, 28A80
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