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VOL. 43 | 2006 Quasi-convex decomposition in o-minimal structures. Application to the gradient conjecture
Krzysztof Kurdyka, Adam Parusiński

Editor(s) Shyuichi Izumiya, Goo Ishikawa, Hiroo Tokunaga, Ichiro Shimada, Takasi Sano


We show that every subset of $\mathbb{R}^n$ definable in an o-minimal structure can be decomposed into a finite number of definable sets that are quasi-convex i.e. have comparable, up to a constant, the intrinsic distance and the distance induced from the embedding. We apply this result to study the limits of secants of the trajectories of gradient vector field $\nabla f$ of a $C^1$ definable function $f$ defined in an open subset of $\mathbb{R}^n$. We show that if the o-minimal structure is polynomially bounded then the limit of such secants exists, that is an analog of the gradient conjecture of R. Thom holds. Moreover we prove that for $n = 2$ the result is true in any o-minimal structure.


Published: 1 January 2006
First available in Project Euclid: 3 January 2019

zbMATH: 1132.32004
MathSciNet: MR2325137

Digital Object Identifier: 10.2969/aspm/04310137

Rights: Copyright © 2006 Mathematical Society of Japan


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