Unbounded components of the singular set of the distance function in ℝⁿ

Abstract

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