Directional properties of sets definable in o-minimal structures
Annales de l'Institut Fourier, Volume 63 (2013) no. 5, pp. 2017-2047.

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 first prove our main theorem and discuss in detail directional properties in the case of an Archimedean real closed field, and in §7 we give a proof in the case of a general real closed field. In addition, related to our main result, we show the existence of special polyhedra in some Euclidean space, illustrating that the bi-Lipschitz equivalence does not always imply the existence of a definable one.

Dans un article précédent par Koike et Paunescu, la notion d’ensemble de directions pour un sous-ensemble d’un espace euclidien a été introduite, et les auteurs ont montré que la dimension de l’ensemble des directions communes de deux sous-ensembles sous-analytiques, nommée la dimension directionnelle, est préservée par un homéomorphisme bi-Lipschitz, à condition que leurs images sont également sous-analytiques. Dans cet article, nous donnons une généralisation de ce résultat à des ensembles définissables dans une structure o-minimale sur un corps réel clos quelconque. Plus précisément, nous prouvons d’abord le théorème principal et nous discussons en détail les propriétés directionnelles dans le cas d’un corps archimèdien réel clos, et dans §7, nous donnons une preuve dans le cas d’un corps général fermé réel. En outre, en relation avec notre résultat principal, nous montrons l’existence des polyèdres spéciaux dans un espace euclidien, ce qui montre que l’équivalence bi-Lipschitz n’implique pas toujours l’existence d’une équivalence définissable.

DOI: 10.5802/aif.2821
Classification: 14P15, 32B20, 14P10, 57R45
Keywords: direction set, o-minimal structure, bi-Lipschitz homeomorphism
Koike, Satoshi 1; Loi, Ta Lê  2; Paunescu, Laurentiu 3; Shiota, Masahiro 4

1 Department of Mathematics, Hyogo University of Teacher Education, Kato, Hyogo 673-1494, Japan
2 Department of Mathematics, University of Dalat, Dalat, Vietnam
3 School of Mathematics, University of Sydney, Sydney, NSW, 2006, Australia
4 Graduate School of Mathematics, Nagoya University, Furo-cho, Chigusa-ku, Nagoya 464-8602, Japan
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     title = {Directional properties of sets definable in~o-minimal structures},
     journal = {Annales de l'Institut Fourier},
     pages = {2017--2047},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {63},
     number = {5},
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Koike, Satoshi; Loi, Ta Lê ; Paunescu, Laurentiu; Shiota, Masahiro. Directional properties of sets definable in o-minimal structures. Annales de l'Institut Fourier, Volume 63 (2013) no. 5, pp. 2017-2047. doi : 10.5802/aif.2821.

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