THE DIRECTIONAL DIMENSION OF SUBANALYTIC SETS IS INVARIANT UNDER BI-LIPSCHITZ HOMEOMORPHISMS

Let A subset of R(n) be a set-germ at 0 is an element of R(n) such that 0 is an element of (A) over bar. We say that r is an element of S(n-1) is a direction of A at 0 is an element of R(n) if there is a sequence of points {x(i)} subset of A \ {0} tending to 0 is an element of R(n) such that x(i)/parallel to x(i)parallel to -> r as i -> infinity. Let D(A) denote the set of all directions of A at 0 is an element of R(n). Let A, B subset of R(n) be subanalytic set-germs at 0 is an element of R(n) such that 0 is an element of (A) over bar boolean AND (B) over bar. We study the problem of whether the dimension of the common direction set, dim(D(A) boolean AND 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.