Directional properties of sets definable in o-minimal structures

TL;DR

This paper generalizes the concept of directional sets and their invariance under bi-Lipschitz maps from subanalytic sets to those definable in any o-minimal structure over real closed fields, highlighting differences in Euclidean polyhedra.

Contribution

It extends the invariance of directional dimension under bi-Lipschitz maps to o-minimal structures over arbitrary real closed fields, broadening previous subanalytic results.

Findings

Directional dimension is preserved under bi-Lipschitz maps in o-minimal structures.

Existence of special polyhedra shows bi-Lipschitz equivalence need not be definable.

Main theorem applies to both Archimedean and general real closed fields.

Abstract

In a former paper the first and third authors introduced the notion of direction set for a subset of R^n, and showed that the dimension of the common direction set of two subanalytic subsets, called directional dimension, is preserved by a bi-Lipschitz homeomorphism, provided that their images are also subanalytic. In this paper we give a generalisation of the above result to sets definable in an o-minimal structure on an arbitrary real closed field. More precisely, we first prove our main theorem and discuss in detail directional properties in the case of an Archimedean real closed field, and then we give a proof in the case of a general real closed field. In addition, related to our main result, we show the existence of special polyhedra in some Euclidean space, illustrating that the bi-Lipschitz equivalence does not always imply the existence of a definable one.