  We prove that Alexandrov's conjecture relating the area and diameter of a
convex surface holds for the surface of a general ellipsoid. This is a direct
consequence of a more general result which estimates the deviation from the
optimal conjectured bound in terms of the length of the cut locus of a point on
the surface. We also prove that the natural extension of the conjecture to
general dimension holds among closed convex spherically symmetric Riemannian
manifolds. Our results are based on a new symmetrization procedure which we
believe to be interesting in its own right.
