  We investigate the maximal open domain $\mathscr{E}(M)$ on which the
orthogonal projection map $p$ onto a subset $M\subseteq \mathbb{R}^d$ can be
defined and study essential properties of $p$. We prove that if $M$ is a $C^1$
submanifold of $\mathbb{R}^d$ satisfying a Lipschitz condition on the tangent
spaces, then $\mathscr{E}(M)$ can be described by a lower semi-continuous
frontier function. We show that this frontier function is continuous if $M$ is
$C^2$ or if the topological skeleton of $M^c$ is closed and we provide an
example showing that the frontier function need not be continuous in general.
  We demonstrate that, for a $C^k$-submanifold $M$ with $k\ge 2$, the
projection map is $C^{k-1}$ on $\mathscr{E}(M)$, and we obtain a
differentiation formula for the projection map which is used to discuss
boundedness of its higher order derivatives on tubular neighborhoods.
  A sufficient condition for the inclusion $M\subseteq\mathscr{E}(M)$ is that
$M$ is a $C^1$ submanifold whose tangent spaces satisfy a local Lipschitz
condition. We prove in a new way that this condition is also necessary. More
precisely, if $M$ is a topological submanifold with $M\subseteq\mathscr{E}(M)$,
then $M$ must be $C^1$ and its tangent spaces satisfy the same local Lipschitz
condition.
  A final section is devoted to highlighting some relations between
$\mathscr{E}(M)$ and the topological skeleton of $M^c$.
