Tangent cones and $C^1$ regularity of definable sets
| Autor: | Kurdyka, Krzysztof, Gal, Olivier Le, Nguyen, Nhan |
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| Rok vydání: | 2017 |
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| Druh dokumentu: | Working Paper |
| Popis: | Let $X\subset \mathbb R^n$ be a connected locally closed definable set in an o-minimal structure. We prove that the following three statements are equivalent: (i) $X$ is a $C^1$ manifold, (ii) the tangent cone and the paratangent cone of $X$ coincide at every point in $X$, (iii) for every $x \in X$, the tangent cone of $X$ at the point $x$ is a $k$-dimensional linear subspace of $\mathbb R^n$ ($k$ does not depend on $x$) varies continuously in $x$, and the density $\theta(X, x) < 3/2$. Comment: 11 pages |
| Databáze: | arXiv |
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