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The directional dimension of subanalytic sets is invariant under bi-Lipschitz homeomorphisms

Satoshi KoikeLaurentiu Paunescu — 2009

Annales de l’institut Fourier

Let A n be a set-germ at 0 n such that 0 A ¯ . We say that r S n - 1 is a direction of A at 0 n if there is a sequence of points { x i } A { 0 } tending to 0 n such that x i x i r as i . Let D ( A ) denote the set of all directions of A at 0 n . Let A , B n be subanalytic set-germs at 0 n such that 0 A ¯ B ¯ . We study the problem of whether the dimension of the common direction set, dim ( D ( A ) D ( B ) ) is preserved by bi-Lipschitz homeomorphisms. We show that although it is not true in general, it is preserved if the images of A and B are also subanalytic. In particular...

Directional properties of sets definable in o-minimal structures

Satoshi KoikeTa Lê LoiLaurentiu PaunescuMasahiro Shiota — 2013

Annales de l’institut Fourier

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 , 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 first prove our main theorem...

Constructing blow-analytic isomorphisms

Toshizumi FukuiTzee-Char KuoLaurentiu Paunescu — 2001

Annales de l’institut Fourier

In this paper we construct non-trivial examples of isomorphisms and we obtain, via toric modifications, an inverse function theorem in this category. We also show that any analytic curve in n , n 3 , can be deformed via a rational blow- analytic isomorphism of n , to a smooth analytic arc.

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