The directional dimension of subanalytic sets is invariant under bi-Lipschitz homeomorphisms
Annales de l'Institut Fourier, Volume 59 (2009) no. 6, p. 2445-2467

Let A n be a set-germ at 0 n such that 0A ¯. We say that rS n-1 is a direction of A at 0 n if there is a sequence of points {x i }A{0} tending to 0 n such that x i x i r as i. Let D(A) denote the set of all directions of A at 0 n .

Let A,B n be subanalytic set-germs at 0 n such that 0A ¯B ¯. We study the problem of whether the dimension of the common direction set, dim(D(A)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.

Soit A n un germe d’ensemble en 0 n tel que 0A ¯. On dit que rS n-1 est une direction de A en 0 n s’il existe une suite de points {x i }A{0} qui converge vers 0 n telle que x i x i r quand i. L’ensemble des directions de A en 0 n est noté D(A). Soient A,B n deux germes en 0 n d’ensemble sous-analytique tels que 0A ¯B ¯.

On étudie le problème suivant : la dimension de l’intersection, dim(D(A)D(B)), est-elle invariante par homéomorphisme bi-Lipschitzien ? On montre que la réponse est non en général, néanmoins la propriété est vraie, lorsque les images de A et B sont sous-analytiques. En particulier, les ensembles des directions de deux germes sous-analytiques, équivalents par homéomorphisme bi-Lipschitzien, ont la même dimension.

DOI : https://doi.org/10.5802/aif.2496
Classification:  14P15,  32B20,  57R45
Keywords: Subanalytic set, direction set, bi-Lipschitz homeomorphism
@article{AIF_2009__59_6_2445_0,
     author = {Koike, Satoshi and Paunescu, Laurentiu},
     title = {The directional dimension of subanalytic sets is invariant under bi-Lipschitz homeomorphisms},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {59},
     number = {6},
     year = {2009},
     pages = {2445-2467},
     doi = {10.5802/aif.2496},
     mrnumber = {2640926},
     zbl = {1184.14086},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2009__59_6_2445_0}
}
Koike, Satoshi; Paunescu, Laurentiu. The directional dimension of subanalytic sets is invariant under bi-Lipschitz homeomorphisms. Annales de l'Institut Fourier, Volume 59 (2009) no. 6, pp. 2445-2467. doi : 10.5802/aif.2496. http://www.numdam.org/item/AIF_2009__59_6_2445_0/

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