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

Cannarsa, Piermarco; Peirone, Roberto

doi:10.1090/S0002-9947-01-02836-7

Unbounded components of the singular set of the distance function in $\mathbb R^n$
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by Piermarco Cannarsa and Roberto Peirone

Trans. Amer. Math. Soc. 353 (2001), 4567-4581

DOI: https://doi.org/10.1090/S0002-9947-01-02836-7

Published electronically: June 1, 2001

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Abstract:

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.

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