  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.
