On gradients of functions definable in o-minimal structures

Krzysztof Kurdyka

Annales de l'institut Fourier (1998)

  • Volume: 48, Issue: 3, page 769-783
  • ISSN: 0373-0956

Abstract

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We prove the o-minimal generalization of the Łojasiewicz inequality ∥ grad f ∥ ≥ | f | α , with α < 1 , in a neighborhood of a , where f is real analytic at a and f ( a ) = 0 . We deduce, as in the analytic case, that trajectories of the gradient of a function definable in an o-minimal structure are of uniformly bounded length. We obtain also that the gradient flow gives a retraction onto levels of such functions.

How to cite

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Kurdyka, Krzysztof. "On gradients of functions definable in o-minimal structures." Annales de l'institut Fourier 48.3 (1998): 769-783. <http://eudml.org/doc/75302>.

@article{Kurdyka1998,
abstract = {We prove the o-minimal generalization of the Łojasiewicz inequality $\Vert \{\rm grad\}\, f\Vert \ge |f|^\alpha $, with $\alpha &lt; 1$, in a neighborhood of $a$, where $f$ is real analytic at $a$ and $f(a)=0$. We deduce, as in the analytic case, that trajectories of the gradient of a function definable in an o-minimal structure are of uniformly bounded length. We obtain also that the gradient flow gives a retraction onto levels of such functions.},
author = {Kurdyka, Krzysztof},
journal = {Annales de l'institut Fourier},
keywords = {flows of gradient; -minimal structure; subanalytic sets; Łojasiewicz inequalities; trajectories of gradient},
language = {eng},
number = {3},
pages = {769-783},
publisher = {Association des Annales de l'Institut Fourier},
title = {On gradients of functions definable in o-minimal structures},
url = {http://eudml.org/doc/75302},
volume = {48},
year = {1998},
}

TY - JOUR
AU - Kurdyka, Krzysztof
TI - On gradients of functions definable in o-minimal structures
JO - Annales de l'institut Fourier
PY - 1998
PB - Association des Annales de l'Institut Fourier
VL - 48
IS - 3
SP - 769
EP - 783
AB - We prove the o-minimal generalization of the Łojasiewicz inequality $\Vert {\rm grad}\, f\Vert \ge |f|^\alpha $, with $\alpha &lt; 1$, in a neighborhood of $a$, where $f$ is real analytic at $a$ and $f(a)=0$. We deduce, as in the analytic case, that trajectories of the gradient of a function definable in an o-minimal structure are of uniformly bounded length. We obtain also that the gradient flow gives a retraction onto levels of such functions.
LA - eng
KW - flows of gradient; -minimal structure; subanalytic sets; Łojasiewicz inequalities; trajectories of gradient
UR - http://eudml.org/doc/75302
ER -

References

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