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Cône normal et régularités de Kuo-Verdier. (Normal cone and Kuo-Verdier regularities). (French) Zbl 1014.58004
Let $$Z$$ be a stratified closed set of $$\mathbb{R}^n$$ whose strata are the $$C^k$$ differentiable varieties, $$k \geq 2$$. If $$Y$$ is an stratum of $$Z$$, the normal cone of $$Z$$ along $$Y$$ consists on the restriction to $$Y$$ of the closure of the set $\{ (x, \mu(x\pi(x))) \mid x \in Z - Y \},$ where $$\pi$$ is the local projection onto $$Y$$ and $$\nu(x) = x / \|x\|$$.
In the paper the authors introduce a new type of Kuo-Verdier regularities, called $$r^e$$, and it is devoted to prove that for an $$(a + r^e)$$ regular $$C^2$$-stratification, ($$(a)$$ being the (a) condition of Whitney), the fiber of the normal cone along a stratum $$Y$$ is equal to the tangent cone of the fiber of a retraction onto $$Y$$. This result generalizes a previous one by the authors which holds for $$(\omega + \delta)$$-differentiable stratifications, $$\omega$$ (respectively, $$\delta$$) being the Kuo-Verdier (respectively, Bekka-Trotman) conditions.

##### MSC:
 58A35 Stratified sets 32S15 Equisingularity (topological and analytic)
##### Keywords:
Kuo-Verdier regularity; stratification; normal cone
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