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The asymmetry of complete and constant width bodies in general normed spaces and the Jung constant

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.