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 thatWe 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.

Bi-Lipschitz homeomorphic subanalytic sets have bi-Lipschitz homeomorphic tangent cones

“…[11], Main Theorem). Let A, B ⊂ R n be subanalytic set-germs at 0 ∈ R n such that 0 ∈ A ∩ B, and h : (R n , 0) → (R n , 0) be a bi-Lipschitz homeomorphism.…”

Directional properties of sets definable in o-minimal structures

“…We give one more example having condition (SSP ). Using a similar argument to Proposition 6.4 in [9], we can show the following: Let us discuss more on the sequence selection property over the field of real numbers R. In this note we consider also the notion of weak sequence selection property, denoted by (W SSP ) for short. Definition 5.9.…”

“…In [9] there are mentioned some directional properties for the original notion of seatangle neighbourhood ST d (A; C). The same properties hold also for our sea-tangle neighbourhood ST θ (A).…”

“…We recall the following lemma. In order to show the above lemma, the following property was used in [9] :…”

“…Therefore we may ask whether the dimension of direction sets is a bi-Lipschitz invariant in the case when the image h(A) is also subanalytic. In fact, the following result is shown in [9]. See H. Hironaka [6] for subanalyticity.…”