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Satoshi Koike; Laurentiu Paunescu
The directional dimension of subanalytic sets is invariant under bi-Lipschitz homeomorphisms
(La dimension directionnelle des ensembles sous-analytiques est invariante par les homéomorphismes bi-Lipschitz)
Annales de l'institut Fourier, 59 no. 6 (2009), p. 2445-2467, doi: 10.5802/aif.2496
Article PDF | Reviews MR 2640926 | Zbl 1184.14086 | 1 citation in Cedram
Class. Math.: 14P15, 32B20, 57R45
Keywords: Subanalytic set, direction set, bi-Lipschitz homeomorphism

Résumé - Abstract

Let $A \subset \mathbb{R}^n$ be a set-germ at $0 \in \mathbb{R}^n$ such that $0 \in \overline{A}$. We say that $r \in S^{n-1}$ is a direction of $A$ at $0 \in \mathbb{R}^n$ if there is a sequence of points $\lbrace x_i \rbrace \subset A \setminus \lbrace 0 \rbrace $ tending to $0 \in \mathbb{R}^n$ such that ${x_i \over \Vert x_i \Vert } \rightarrow r$ as $i \rightarrow \infty $. Let $D(A)$ denote the set of all directions of $A$ at $0 \in \mathbb{R}^n$.

Let $A, \ B \subset \mathbb{R}^n$ be subanalytic set-germs at $0 \in \mathbb{R}^n$ such that $0 \in \overline{A} \cap \overline{B}$. We study the problem of whether the dimension of the common direction set, $\dim (D(A) \cap D(B))$ is preserved by bi-Lipschitz homeomorphisms. We show that although it is not true in general, it is preserved if the images of $A$ and $B$ are also subanalytic. In particular if two subanalytic set-germs are bi-Lipschitz equivalent their direction sets must have the same dimension.

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