Dimension de Hausdorff de la Nervure

International audienceFor 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 $\mathrm{dim_H}(\mathcal{N}\setminus \mathcal{M})=1$