2018/11/26 by Gunther Leobacher, Leobacher, Gunther, Alexander Steinicke +1 · 3 citations
Computer Science · Mathematics · #53A07 #57N40 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Optimization and Variational Analysis #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.1811.10578
openalex publication_date 2018/11/26 · openalex created_date 2022/08/01 · openalex updated_date 2026/07/28
We investigate the maximal open domain mathscrE(M) on which the\northogonal projection map p onto a subset M\⊆ \ℝd can be\ndefined and study essential properties of p. We prove that if M is a C1\nsubmanifold of \ℝd satisfying a Lipschitz condition on the tangent\nspaces, then mathscrE(M) can be described by a lower semi-continuous\nfrontier function. We show that this frontier function is continuous if M is\nC2 or if the topological skeleton of Mc is closed and we provide an\nexample showing that the frontier function need not be continuous in general.\n We demonstrate that, for a Ck-submanifold M with k\≥ 2, the\nprojection map is Ck-1 on mathscrE(M), and we obtain a\ndifferentiation formula for the projection map which is used to discuss\nboundedness of its higher order derivatives on tubular neighborhoods.\n A sufficient condition for the inclusion M\⊆ mathscrE(M) is that\nM is a C1 submanifold whose tangent spaces satisfy a local Lipschitz\ncondition. We prove in a new way that this condition is also necessary. More\nprecisely, if M is a topological submanifold with M\⊆ mathscrE(M),\nthen M must be C1 and its tangent spaces satisfy the same local Lipschitz\ncondition.\n A final section is devoted to highlighting some relations between\n mathscrE(M) and the topological skeleton of Mc.\n