Unbounded components of the singular set of the distance function in \(\mathbb{R}^n\). (English) Zbl 1021.49013
Summary: 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.
MSC:
| 49J52 | Nonsmooth analysis |
| 34A60 | Ordinary differential inclusions |
| 49J53 | Set-valued and variational analysis |
| 41A52 | Uniqueness of best approximation |
| 41A65 | Abstract approximation theory (approximation in normed linear spaces and other abstract spaces) |
Keywords:
distance function; metric projection; best approximation; singularities; differential inclusions; Motzkin theoremReferences:
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