Directional properties of sets definable in o-minimal structures

Satoshi Koike[1]; Ta Lê Loi[2]; Laurentiu Paunescu[3]; Masahiro Shiota[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

Annales de l’institut Fourier (2013)

  • Volume: 63, Issue: 5, page 2017-2047
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Koike, Satoshi, et al. "Directional properties of sets definable in o-minimal structures." Annales de l’institut Fourier 63.5 (2013): 2017-2047. <http://eudml.org/doc/275620>.

@article{Koike2013,
abstract = {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.},
affiliation = {Department of Mathematics, Hyogo University of Teacher Education, Kato, Hyogo 673-1494, Japan; Department of Mathematics, University of Dalat, Dalat, Vietnam; School of Mathematics, University of Sydney, Sydney, NSW, 2006, Australia; Graduate School of Mathematics, Nagoya University, Furo-cho, Chigusa-ku, Nagoya 464-8602, Japan},
author = {Koike, Satoshi, Loi, Ta Lê, Paunescu, Laurentiu, Shiota, Masahiro},
journal = {Annales de l’institut Fourier},
keywords = {direction set; o-minimal structure; bi-Lipschitz homeomorphism},
language = {eng},
number = {5},
pages = {2017-2047},
publisher = {Association des Annales de l’institut Fourier},
title = {Directional properties of sets definable in o-minimal structures},
url = {http://eudml.org/doc/275620},
volume = {63},
year = {2013},
}

TY - JOUR
AU - Koike, Satoshi
AU - Loi, Ta Lê
AU - Paunescu, Laurentiu
AU - Shiota, Masahiro
TI - Directional properties of sets definable in o-minimal structures
JO - Annales de l’institut Fourier
PY - 2013
PB - Association des Annales de l’institut Fourier
VL - 63
IS - 5
SP - 2017
EP - 2047
AB - 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.
LA - eng
KW - direction set; o-minimal structure; bi-Lipschitz homeomorphism
UR - http://eudml.org/doc/275620
ER -

References

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