Abstract
Let v 1 , … , v m be a finite set of unit vectors in R n . Suppose that an infinite sequence of Steiner symmetrizations are applied to a compact convex set K in R 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.
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