  A subset of a convex body $B$ containing the origin in a Euclidean space is
{\it parkable in $B$} if it can be translated inside $B$ in such a manner that
the translate the origin. We provide characterizations of ellipsoids and of
centrally symmetric convex bodies in Euclidean spaces of dimension $\ge 3$
based on the notion of parkability, answering several questions posed by G.
Bergman.
  The techniques used, which are based on characterizations of Hilbert spaces
among finite-dimensional Banach spaces in terms of their lattices of subspaces
and algebras of endomorphisms, also apply to improve a result of W. Blaschke
characterizing ellipsoids in terms of boundaries of illumination.
