Existence, uniqueness and regularity of the projection onto differentiable manifolds

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Existence, uniqueness and regularity of the projection onto differentiable manifolds. / Leobacher, Gunther; Steinicke, Alexander.
in: Annals of global analysis and geometry, Jahrgang 60.2021, Nr. 3, 10.2021, S. 559-587.

Publikationen: Beitrag in FachzeitschriftArtikelForschung(peer-reviewed)

Vancouver

Leobacher G, Steinicke A. Existence, uniqueness and regularity of the projection onto differentiable manifolds. Annals of global analysis and geometry. 2021 Okt;60.2021(3):559-587. Epub 2021 Jul 1. doi: 10.1007/s10455-021-09788-z

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@article{c82be3365e424a28ab5ce7c1e93d0788,
title = "Existence, uniqueness and regularity of the projection onto differentiable manifolds",
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. ",
keywords = "Nonlinear orthogonal projection, Medial axis, Sets of positive reach, Tubular neighborhood",
author = "Gunther Leobacher and Alexander Steinicke",
note = "Publisher Copyright:{\textcopyright} 2021, The Author(s).",
year = "2021",
month = oct,
doi = "10.1007/s10455-021-09788-z",
language = "English",
volume = "60.2021",
pages = "559--587",
journal = " Annals of global analysis and geometry",
issn = "0232-704x",
publisher = "Springer Netherlands",
number = "3",

}

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TY - JOUR

T1 - Existence, uniqueness and regularity of the projection onto differentiable manifolds

AU - Leobacher, Gunther

AU - Steinicke, Alexander

N1 - Publisher Copyright:© 2021, The Author(s).

PY - 2021/10

Y1 - 2021/10

N2 - 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.

AB - 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.

KW - Nonlinear orthogonal projection

KW - Medial axis

KW - Sets of positive reach

KW - Tubular neighborhood

UR - http://www.scopus.com/inward/record.url?scp=85114056928&partnerID=8YFLogxK

U2 - 10.1007/s10455-021-09788-z

DO - 10.1007/s10455-021-09788-z

M3 - Article

VL - 60.2021

SP - 559

EP - 587

JO - Annals of global analysis and geometry

JF - Annals of global analysis and geometry

SN - 0232-704x

IS - 3

ER -