## 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}.

Original language | English |
---|---|

Pages (from-to) | 559-587 |

Number of pages | 29 |

Journal | Annals of global analysis and geometry |

Volume | 60.2021 |

Issue number | 3 |

Early online date | 1 Jul 2021 |

DOIs | |

Publication status | Published - Oct 2021 |

### Bibliographical note

Publisher Copyright:© 2021, The Author(s).## Keywords

- Nonlinear orthogonal projection
- Medial axis
- Sets of positive reach
- Tubular neighborhood