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.Comment: 18 pages. The essential results are the same as in the previous version. This version includes a more thorough introduction, some clarifications in the proofs, an updated bibliography, and some open questions at the en
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