The Lipschitz type of the geometric directional bundle (Q6564860)

The authors introduced two notions in their previous studies. The first one is the concept of sequence selection property (SSP) introduced in [\textit{S. Koike} and \textit{L. Paunescu}, J. Math. Soc. Japan 67(2), 721--751 (2015; Zbl 1326.14137)]. The other one is the geometric directional bundle of a subset of \(\mathbb R^n\) in [\textit{S. Koike} and \textit{L. Paunescu}, Rev. Roum. Math. Pures Appl. 64(4), 479--501 (2019; Zbl 1463.14007)]. In this paper, the authors consider the problem whether these two are preserved under bi-Lipchitz equivalence. Both positive and negative results are presented. Theorem 5.4 asserts that the image of a subanalytic set under a bi-Lipchitz homeomorphism enjoys SSP on a dense set. A sufficient condition for the geometric directional bundle to be preserved under a bi-Lipchitz homeomorphism is given in Theorem 4.7, but it is not preserved in multiple examples. For instance, an example illustrates that the geometric directional bundle is not preserved even under everywhere differentiable bi-Lipchitz homeomorphism.