Abstract
This paper is devoted to the study of the medial axes of sets definable in polynomially bounded o-minimal structures, i.e. the sets of points with more than one closest point with respect to the Euclidean distance. Our point of view is that of singularity theory. While trying to make the paper self-contained, we gather here also a large bunch of basic results. Our main interest, however, goes to the characterization of those singular points of a definable, closed set Xsubset {mathbb {R}}^n, which are reached by the medial axis.
Highlights
1 Introduction the present paper is essentially concerned with sets definable in some ominimal structure that in addition should be polynomially bounded, many of the results presented hold true for subanalytic sets, but sometimes even in general
Throughout this paper, definable means definable in some polynomially bounded ominimal structure expanding the field of reals R
The relation between the medial axis and the central set is known and given in the following theorem which we prove in a different way than it is done in [13]
Summary
The present paper is essentially concerned with sets definable in some ominimal structure that in addition should be polynomially bounded, many of the results presented hold true for subanalytic sets (which—recall—do not form an o-minimal structure), but sometimes even in general. This is clearly apparent from the proofs, and we will not stress this fact. Despite the fact that the subject of medial axes, central sets, skeletons, cut loci (all these notions denote more or less the same concept and are often incorrectly interchanged) has been extensively studied, astonishingly few results concern the relations to singularities. The bibliography we present is certainly far from being exhaustive
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