Dimension de Hausdorff de la Nervure

Rivière, Alain

HAL, hal-00341527 / Harvested from HAL

Résumé

For a separable Hilbert space $E$ whose dimension is $\geq 2$ and for an open subset $\Omega$ of $E$, not empy and different from $E$, let $\mathcal{M}$ be the set of all points of $\Omega$ which have at least two projections on the close set $E \setminus \Omega$, and let $\mathcal{N}$ be the set of all the centres of the open balls contained in $\Omega$ and which are maximal for inclusion. We show that the Hausdorff dimension $\mathrm{dim_H}(\mathcal{N} \setminus \mathcal{M})$ of $\mathcal{N} \setminus \mathcal{M}$ may be any real value $s$ such that $0 \leq s \leq \dim E$; we also show that $\Omega$ can be chosen so that $\mathcal{N}$ is everywhere dense in $\Omega$ and so that we have $\mathrm{dim_H}(\mathcal{N}\setminus \mathcal{M})=1$.

Détail

Publié le : 2001-07-05
Classification:  cut locus,  Hausdorff dimension,  set-valued function,  MSC: 28A78, 28A80, 46C05,  [MATH.MATH-MG]Mathematics [math]/Metric Geometry [math.MG]