## Abstract

We investigate the maximal open domain E(M) on which the orthogonal projection map p onto a subset M⊆ R
^{d} can be defined and study essential properties of p. We prove that if M is a C
^{1} submanifold of R
^{d} satisfying a Lipschitz condition on the tangent spaces, then E(M) can be described by a lower semi-continuous function, named frontier function. We show that this frontier function is continuous if M is C
^{2} or if the topological skeleton of M
^{c} is closed and we provide an example showing that the frontier function need not be continuous in general. We demonstrate that, for a C
^{k}-submanifold M with k≥ 2 , the projection map is C
^{k}
^{-}
^{1} on E(M) , and we obtain a differentiation formula for the projection map which is used to discuss boundedness of its higher order differentials on tubular neighborhoods. A sufficient condition for the inclusion M⊆ E(M) is that M is a C
^{1} submanifold whose tangent spaces satisfy a local Lipschitz condition. We prove in a new way that this condition is also necessary. More precisely, if M is a topological submanifold with M⊆ E(M) , then M must be C
^{1} and its tangent spaces satisfy the same local Lipschitz condition. A final section is devoted to highlighting some relations between E(M) and the topological skeleton of M
^{c}.

Originalsprache | Englisch |
---|---|

Seiten (von - bis) | 559-587 |

Seitenumfang | 29 |

Fachzeitschrift | Annals of global analysis and geometry |

Jahrgang | 60.2021 |

Ausgabenummer | 3 |

Frühes Online-Datum | 1 Juli 2021 |

DOIs | |

Publikationsstatus | Veröffentlicht - Okt. 2021 |

### Bibliographische Notiz

Funding Information:Open access funding provided by University of Graz. Gunther Leobacher and Alexander Steinicke are supported by the Austrian Science Fund (FWF): Project F5508-N26, which is part of the Special Research Program ‘Quasi-Monte Carlo Methods: Theory and Applications’.

Publisher Copyright:

© 2021, The Author(s).