  We give a systematic and thorough study of geometric notions and results
connected to Minkowski's measure of symmetry and the extension of the
well-known Minkowski functional to arbitrary, not necessarily symmetric convex
bodies K on any (real) normed space X. Although many of the notions and results
we treat in this paper can be found elsewhere in the literature, they are
scattered and possibly hard to find. Further, we are not aware of a systematic
study of this kind and we feel that several features, connections and
properties - e.g. the connections between many equivalent formulations - are
new, more general and they are put in a better perspective now. In particular,
we prove a number of fundamental properties of the extended Minkowski
functional, including convexity, global Lipschitz boundedness, linear growth
and approximation of the classical Minkowski functional of the central
symmetrization of the body K.
  Our aim is to present how in the recent years these notions proved to be
surprisingly relevant and effective in problems of approximation theory.
