  Let $v_1, ..., v_m$ be a finite set of unit vectors in $\RR^n$. Suppose that
an infinite sequence of Steiner symmetrizations are applied to a compact convex
set $K$ in $\RR^n$, where each of the symmetrizations is taken with respect to
a direction from among the $v_i$. Then the resulting sequence of Steiner
symmetrals always converges, and the limiting body is symmetric under
reflection in any of the directions $v_i$ that appear infinitely often in the
sequence. In particular, an infinite periodic sequence of Steiner
symmetrizations always converges, and the set functional determined by this
infinite process is always idempotent.

Metric Geometry