Steiner symmetrization using a finite set of directions

Let v1,…,vm be a finite set of unit vectors in Rn. Suppose that an infinite sequence of Steiner symmetrizations are applied to a compact convex set K in Rn, where each of the symmetrizations is taken with respect to a direction from among the vi. Then the resulting sequence of Steiner symmetrals always converges, and the limiting body is symmetric under reflection in any of the directions vi 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.