# The directional dimension of subanalytic sets is invariant under bi-Lipschitz homeomorphisms

• [1] Hyogo University of Teacher Education Department of Mathematics Kato, Hyogo 673-1494 (Japan)
• [2] University of Sydney School of Mathematics and Statistics Sydney, NSW, 2006 (Australia)
• Volume: 59, Issue: 6, page 2445-2467
• ISSN: 0373-0956

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## Abstract

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Let $A\subset {ℝ}^{n}$ be a set-germ at $0\in {ℝ}^{n}$ such that $0\in \overline{A}$. We say that $r\in {S}^{n-1}$ is a direction of $A$ at $0\in {ℝ}^{n}$ if there is a sequence of points $\left\{{x}_{i}\right\}\subset A\setminus \left\{0\right\}$ tending to $0\in {ℝ}^{n}$ such that $\frac{{x}_{i}}{\parallel {x}_{i}\parallel }\to r$ as $i\to \infty$. Let $D\left(A\right)$ denote the set of all directions of $A$ at $0\in {ℝ}^{n}$.Let $A,\phantom{\rule{4pt}{0ex}}B\subset {ℝ}^{n}$ be subanalytic set-germs at $0\in {ℝ}^{n}$ such that $0\in \overline{A}\cap \overline{B}$. We study the problem of whether the dimension of the common direction set, $dim\left(D\left(A\right)\cap D\left(B\right)\right)$ 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.

## How to cite

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Koike, Satoshi, and Paunescu, Laurentiu. "The directional dimension of subanalytic sets is invariant under bi-Lipschitz homeomorphisms." Annales de l’institut Fourier 59.6 (2009): 2445-2467. <http://eudml.org/doc/10460>.

@article{Koike2009,
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.},
affiliation = {Hyogo University of Teacher Education Department of Mathematics Kato, Hyogo 673-1494 (Japan); University of Sydney School of Mathematics and Statistics Sydney, NSW, 2006 (Australia)},
author = {Koike, Satoshi, Paunescu, Laurentiu},
journal = {Annales de l’institut Fourier},
keywords = {Subanalytic set; direction set; bi-Lipschitz homeomorphism; subanalytic set; invariant of a subanalytic set},
language = {eng},
number = {6},
pages = {2445-2467},
publisher = {Association des Annales de l’institut Fourier},
title = {The directional dimension of subanalytic sets is invariant under bi-Lipschitz homeomorphisms},
url = {http://eudml.org/doc/10460},
volume = {59},
year = {2009},
}

TY - JOUR
AU - Koike, Satoshi
AU - Paunescu, Laurentiu
TI - The directional dimension of subanalytic sets is invariant under bi-Lipschitz homeomorphisms
JO - Annales de l’institut Fourier
PY - 2009
PB - Association des Annales de l’institut Fourier
VL - 59
IS - 6
SP - 2445
EP - 2467
AB - 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.
LA - eng
KW - Subanalytic set; direction set; bi-Lipschitz homeomorphism; subanalytic set; invariant of a subanalytic set
UR - http://eudml.org/doc/10460
ER -

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