The Kuratowski convergence of medial axes and conflict sets

This paper consists of two parts. In the first one we study the behaviour of medial axes (skeletons) of closed, definable (in some o-minimal structure) sets in {\Rz}^n under deformations. The second one is devoted to a similar study of conflict sets in definable families. We apply a new approach to the deformation process. Instead of seeing it as a `jump' from the initial to the final state, we perceive it as a continuous process, expressed using the Kuratowski convergence of sets (hence, unlike other authors, we do not require any regularity of the deformation). Our main `medial axis inner semi-continuity' result has already proved useful, as it was used to compute the tangent cone of the medial axis with application in singularity theory.Comment: The preprint has been extended to include also the study of the behaviour of the conflict set of a continuous family of definable sets performed with a new co-author. Therefore the title has slightly been changed, too. Besides that, the references have also been updated and in the last version we strengthened the statement of Theorem 5.1

A long and winding road to definable sets

We survey the development of o-minimal structures from a geometric point of view and compare them with subanalytic sets insisting on the differences. The idea is to show the long way from semi-analytic to definable sets, from normal partitions to cell decompositions. Some recent results are discussed in the last section