Hausdorff Dimension of Cut Loci of Generic Subspaces of Euclidean Spaces

\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 $\mathbb{E}^d$.