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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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


Authors: Piermarco Cannarsa and Roberto Peirone
Journal: Trans. Amer. Math. Soc. 353 (2001), 4567-4581
MSC (1991): Primary 41A65, 26A27; Secondary 34A60, 49J52
Published electronically: June 1, 2001
MathSciNet review: 1851184
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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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Additional Information

Piermarco Cannarsa
Affiliation: Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica, 00133 Roma (Italy)
Email: cannarsa@ mat.uniroma2.it

Roberto Peirone
Affiliation: Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica, 00133 Roma (Italy)
Email: peirone@ mat.uniroma2.it

DOI: http://dx.doi.org/10.1090/S0002-9947-01-02836-7
PII: S 0002-9947(01)02836-7
Keywords: Distance function, metric projection, best approximation, singularities, differential inclusions
Received by editor(s): October 19, 2000
Received by editor(s) in revised form: December 20, 2000
Published electronically: June 1, 2001
Article copyright: © Copyright 2001 American Mathematical Society