Stabilisation of geometric directional bundle for a subanalytic set

In the previous paper [8] we have introduced the notion of geometric directional bundle of a singular space, in order to introduce global bi-Lipschitz invariants. Then we have posed the question of whether or not the geometric directional bundle is stabilised as an operation acting on singular spaces. In this paper we give a positive answer in the case where the singular spaces are subanalytic sets, thus providing a new invariant associated with the subanalytic sets.(c) 2021 Elsevier B.V. All rights reserved.