  Steiner and Schwarz symmetrizations, and their most important relatives, the
Minkowski, Minkowski-Blaschke, fiber, inner rotational, and outer rotational
symmetrizations, are investigated. The focus is on the convergence of
successive symmetrals with respect to a sequence of $i$-dimensional subspaces
of $\mathbb{R}^n$. Such a sequence is called universal for a family of sets if
the successive symmetrals of any set in the family converge to a ball with
center at the origin. New universal sequences for the main symmetrizations, for
all valid dimensions $i$ of the subspaces, are found, by combining two groups
of results. The first, published separately, provides finite sets
${\mathcal{F}}$ of subspaces such that reflection symmetry (or rotational
symmetry) with respect to each subspace in ${\mathcal{F}}$ implies full
rotational symmetry. In the second, proved here, a theorem of Klain for Steiner
symmetrization is extended to Schwarz, Minkowski, Minkowski-Blaschke, and fiber
symmetrizations, showing that if a sequence of subspaces is drawn from a finite
set ${\mathcal{F}}$ of subspaces, the successive symmetrals of any compact
convex set converge to a compact convex set that is symmetric with respect to
any subspace in ${\mathcal{F}}$ appearing infinitely often in the sequence. It
is also proved that for Steiner, Schwarz, and Minkowski symmetrizations, a
sequence of $i$-dimensional subspaces is universal for the class of compact
sets if and only if it is universal for the class of compact convex sets, and
Klain's theorem is shown to hold for Schwarz symmetrization of compact sets.
