Unbounded components of the singular set of the distance function in $\mathbb R^n$

Given a closed set $F\subseteq \mathbb{R}^{n}$, the set $\Sigma _{F}$ of all points at which the metric projection onto $F$ is multi-valued is nonempty if and only if $F$ is nonconvex. The authors analyze such a set, characterizing the unbounded connected components of $\Sigma _{F}$. For $F$ compact, the existence of an asymptote for any unbounded component of $\Sigma _{F}$ is obtained.