The main purpose of this paper is to present a natural method of decomposition into special cubes and to demonstrate how it makes it possible to efficiently achieve many well-known fundamental results from quasianalytic geometry as, for instance, Gabrielov's complement theorem, o-minimality or quasianalytic cell decomposition.
We prove that for a finite collection of sets definable in an o-minimal structure there exists a compatible definable stratification such that for any stratum the fibers of its projection onto satisfy the Whitney property with exponent 1.
It is shown that a sub-analytic set has a density at each point, and the notion of pure cone is defined. As in the complex case, this density may be expressed in terms of the area of the connected components of the pure tangent cone, with involved integral multiplicities.
We present a tameness property of sets definable in o-minimal structures by showing that Morse functions on a definable closed set form a dense and open subset in the space of definable functions endowed with the Whitney topology.
In a previous paper by Koike and Paunescu, it was introduced the notion of direction set for a subset of a Euclidean space, and it was shown that the dimension of the common direction set of two subanalytic subsets, called the directional dimension, is preserved by a bi-Lipschitz homeomorphism, provided that their images are also subanalytic. In this paper we give a generalisation of the above result to sets definable in an o-minimal structure on an arbitrary real closed field. More precisely, we...
Pour un ensemble sous-analytique, connexe fermé, la distance géodésique est atteinte et est uniformément équivalente, avec des constantes arbitrairement proches de 1, à une distance sous-analytique.