Existence, uniqueness and regularity of the projection onto differentiable manifolds
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in: Annals of global analysis and geometry, Jahrgang 60.2021, Nr. 3, 10.2021, S. 559-587.
Publikationen: Beitrag in Fachzeitschrift › Artikel › Forschung › (peer-reviewed)
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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 -