  It is a classical fact, that given an arbitrary n-dimensional convex body,
there exists an appropriate sequence of Minkowski symmetrizations (or Steiner
symmetrizations), that converges in Hausdorff metric to a Euclidean ball. Here
we provide quantitative estimates regarding this convergence, for both
Minkowski and Steiner symmetrizations. Our estimates are polynomial in the
dimension and in the logarithm of the desired distance to a Euclidean ball,
improving previously known exponential estimates. Inspired by a method of
Diaconis, our technique involves spherical harmonics. We also make use of an
earlier result by the author regarding ``isomorphic Minkowski symmetrization''.
