82 research outputs found
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
Ensembles sous-analytiques à la polonaise: avec une introduction aux fonctions et ensembles analytiques
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