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2018 Strict $C^1$-triangulations in o-minimal structures
Małgorzata Czapla, Wiesław Pawłucki
Topol. Methods Nonlinear Anal. 52(2): 739-747 (2018). DOI: 10.12775/TMNA.2018.033


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]).


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Małgorzata Czapla. Wiesław Pawłucki. "Strict $C^1$-triangulations in o-minimal structures." Topol. Methods Nonlinear Anal. 52 (2) 739 - 747, 2018.


Published: 2018
First available in Project Euclid: 25 July 2018

zbMATH: 07051690
MathSciNet: MR3915661
Digital Object Identifier: 10.12775/TMNA.2018.033

Rights: Copyright © 2018 Juliusz P. Schauder Centre for Nonlinear Studies


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Vol.52 • No. 2 • 2018
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