On Lipschitz Geometry at infinity of complex analytic sets

Abstract

In this article, we prove that for a definable set in an ominimal 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 stereographic modifications are outer lipeomorphic if and only if their inversions are outer lipeomorphic.