Steiner symmetrization using a finite set of directions

Abstract

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.

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 end