Nervure d'un Ouvert d'un Espace Euclidien
Résumé
We study in this work the set $\mathcal{N}$ of the points of a bounded and connected open subset $\Omega$ of a Euclidean space $\mathcal{E}$, which doesn't belong to the interior of any segment joining a point of $\Omega$ to one of its projections to the complement $\mathcal{E} \setminus \Omega$ of $\Omega$. For instance the points of $\Omega$ which have many projections to $\mathcal{E} \setminus \Omega$ are in $\mathcal{N}$; we begin by showing that the set $\mathcal{M}$ of these last ones can be dense in $\Omega$, but is included in a countable union of Lipschitzian submanifolds of $\mathcal{E}$; then we introduce $\mathcal{N}$ and we prove that it is Lebesgue negligible. The study of a relation defined by convexity properties provide us with a fundamental instrument which we use first for showing that $\mathcal{M}$ and $\mathcal{N}$ are locally connected and that they have the same homotopy type as $\Omega$, then for describing trajectories of a mobile point which at every moment tries to move away as quickly as possible from $\mathcal{E} \setminus \Omega$ of $\Omega$.