## Advanced Studies in Pure Mathematics

- Adv. Stud. Pure Math.
- Singularity Theory and Its Applications, S. Izumiya, G. Ishikawa, H. Tokunaga, I. Shimada and T. Sano, eds. (Tokyo: Mathematical Society of Japan, 2006), 137 - 177

### Quasi-convex decomposition in o-minimal structures. Application to the gradient conjecture

Krzysztof Kurdyka and Adam Parusiński

#### Abstract

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.

#### Article information

**Dates**

Received: 8 June 2004

Revised: 27 June 2005

First available in Project Euclid:
3 January 2019

**Permanent link to this document**

https://projecteuclid.org/
euclid.aspm/1546543235

**Digital Object Identifier**

doi:10.2969/aspm/04310137

**Mathematical Reviews number (MathSciNet)**

MR2325137

**Zentralblatt MATH identifier**

1132.32004

#### Citation

Kurdyka, Krzysztof; Parusiński, Adam. Quasi-convex decomposition in o-minimal structures. Application to the gradient conjecture. Singularity Theory and Its Applications, 137--177, Mathematical Society of Japan, Tokyo, Japan, 2006. doi:10.2969/aspm/04310137. https://projecteuclid.org/euclid.aspm/1546543235