  In this paper we state a one-to-one connection between the maximal ratio of
the circumradius and the diameter of a body (the Jung constant) in an arbitrary
Minkowski space and the maximal Minkowski asymmetry of the complete bodies
within that space. This allows to generalize and unify recent results on
complete bodies and to derive a necessary condition on the unit ball of the
space, assuming a given body to be complete. Finally, we state several
corollaries, i.e. concerning the Helly dimension or the Banach-Mazur distance.
