## Topological Methods in Nonlinear Analysis

### Strict $C^1$-triangulations in o-minimal structures

#### Abstract

Inspired by the recent articles of T. Ohmoto and M. Shiota [9], [10] on $C^1$-triangulations of semialgebraic sets, we prove here by using different methods the following theorem: Let $R$ be a real closed field and let an expansion of $R$ to an o-minimal structure be given. Then for any closed bounded definable subset $A$ of $R^n$ and a finite family $B_1,\dots,B_r$ of definable subsets of $A$ there exists a definable triangulation $h\colon |\mathcal K|\rightarrow A$ of $A$ compatible with $B_1,\dots,B_r$ such that $\mathcal K$ is a simplicial complex in $R^n$, $h$ is a $C^1$-embedding of each (open) simplex $\Delta\in \mathcal K$ and $h$ extends to a definable $C^1$-mapping defined on a definable open neighborhood of $|\mathcal K|$ in $R^n$. This improves Ohmoto-Shiota's theorem in three ways; firstly, $h$ is a $C^1$-embedding on each simplex; secondly, the simplicial complex $\mathcal K$ is in the same space as $A$ and thirdly, our proof is performed for any o-minimal structure. The possibility to have $h$ with the first of these properties was stated by Ohmoto and Shiota as an open problem (see [9]).

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 52, Number 2 (2018), 739-747.

Dates
First available in Project Euclid: 25 July 2018

https://projecteuclid.org/euclid.tmna/1532484293

Digital Object Identifier
doi:10.12775/TMNA.2018.033

Mathematical Reviews number (MathSciNet)
MR3915661

Zentralblatt MATH identifier
07051690

#### Citation

Czapla, Małgorzata; Pawłucki, Wiesław. Strict $C^1$-triangulations in o-minimal structures. Topol. Methods Nonlinear Anal. 52 (2018), no. 2, 739--747. doi:10.12775/TMNA.2018.033. https://projecteuclid.org/euclid.tmna/1532484293

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