2010/03/01 by Satoshi Koike, Ta Lê Loi, Koike, Satoshi +5 · 1 citation
Computer Science · Mathematics · #14P15 #Advanced Topology and Set Theory #Algebraic Geometry (math.AG) #Computability, Logic, AI Algorithms #FOS: Mathematics #Mathematical and Theoretical Analysis
paper · pdf · doi:10.48550/arxiv.1003.0244
openalex publication_date 2010/03/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In a former paper the first and third authors introduced the notion of direction set for a subset of Rn, 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.