The complex gradient inequality with parameter

We prove that given a holomorphic family of holomorphic functions with isolated singularities at zero and constant Milnor number, it is possible to obtain the gradient inequality with a uniform exponent.Comment: A remark was added at the end and some misprints were correcte

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

On the points realizing the distance to a definable set

AbstractWe prove a definable/subanalytic version of a useful lemma, presumably due to John Nash, concerning the points realizing the Euclidean distance to an analytic submanifold of Rn. We present a parameter version of the main result and we discuss the properties of the multifunction obtained