Local vs. global Lipschitz geometry ...

• Main Author: Sampaio, José Edson
• Format: Report
• Language: unknown
• Published: arXiv 2023
• Subjects: Metric Geometry math.MG; Algebraic Geometry math.AG; Logic math.LO; FOS Mathematics; 14Pxx, 53Cxx primary, 32S50 secondary; North Pole
• Online Access: https://dx.doi.org/10.48550/arxiv.2305.11830; https://arxiv.org/abs/2305.11830

In this article, we prove that for a definable set in an o-minimal structure with connected link (at 0 or infinity), the inner distance of the link is equivalent to the inner distance of the set restrict to the link. As consequences, we obtain: (1) a definable set, with connected link at infinity, is LNE at infinity if and only if it is LLNE at infinity; (2) a definable set is LNE at infinity if and only if its stereographic modification is LNE at the North Pole; (3) a connected definable set is LNE if and only if its stereographic modification is LNE; and under certain extra conditions we prove that: (4) two definable sets are definably inner (resp. outer) lipeomorphic if and only if their stereographic modifications are definably inner (resp. outer) lipeomorphic if and only if their inversions are definably inner (resp. outer) lipeomorphic. Moreover, we also prove that two sets in Euclidean spaces, not necessarily definables in an o-minimal structure, are outer lipeomorphic if and only if their ... : 16 pages and 1 figure ...