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

Abstract

Let A ⊂ R n be a set-germ at 0 ∈ R n such that 0 ∈ A. We say that r ∈ S n−1 is a direction of A at 0 ∈ R n if there is a sequence of points {xi} ⊂ A\{0} tending to 0 ∈ R n such that x i k xik → r as i → ∞. Let D(A) denote the set of all directions of A at 0 ∈ R n . Let A, B ⊂ R n be subanalytic set-germs at 0 ∈ R n such that 0 ∈ A ∩ 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. Let us recall the notion of blow-analyticity. Let f, g : (R n ,0) → (R,0) be analytic function-germs. We say that they are blow-analytically equivalent if there are real modifications � : (M, � −1 (0)) → (R n ,0), � ' : (M ' , � '−1 (0)) → (R n ,0) and an analytic isomorphism � : (M, � −1 (0)) → (M ' , � '−1 (0)) which induces a homeomorphism φ : (R n ,0) → (R n ,0), � ' ◦ � = φ ◦ � , such that f = g ◦ φ. A blow-analytic homeomorphism is such a φ, a homeomorphism induced by an analytic isomorphism via real modifications.