2025/12/19 by Jan Rataj, Rataj, Jan, Luděk Zaj́ıček +1
Mathematics · #Point processes and geometric inequalities #Nonlinear Partial Differential Equations #Limits and Structures in Graph Theory
paper · doi:10.48550/arxiv.2512.17606
The main result of the article is a complete characterization of the local structure of two-dimensional sets with positive reach in Rd. We also present a more elementary proof of a recent result of A. Lytchak which describes for k≤ d the local structure of k-dimensional sets with positive reach A in Rd at points where the tangent cone of A is k-dimensional. As an easy corollary of our and Lytchak's results we obtain a characterization of compact two-dimensional sets with positive reach in Rd. Our method also shows that, for any set A⊂ Rd with positive reach, the set of points at which the tangent cone of A is k-dimensional is locally contained in a k-dimensional C1,1 surface. As a consequence we obtain that if 1≤ k