  We study spaces with directionally asymptotically controlled ellipsoids
approximating the unit ball in finite-dimensions. These ellipsoids are the
unique minimum volume ellipsoids, which contain the unit ball of the
corresponding finite-dimensional subspace. The directional control here means
that we evaluate the ellipsoids with a given functional of the dual space. The
term asymptotical refers to the fact that we take '$\limsup$' over
finite-dimensional subspaces.
  This leads to some isomorphic and isometric characterizations of Hilbert
spaces. An application involving Mazur's rotation problem is given. We also
discuss the complexity of the family of ellipsoids as the dimension and
geometry vary.
