Steiner symmetrization using a finite set of directions

TL;DR

This paper proves that applying Steiner symmetrizations in a finite set of directions repeatedly to a convex set always results in convergence to a symmetric shape, with the limit reflecting the directions used infinitely often.

Contribution

It establishes convergence and symmetry properties of Steiner symmetrizations when using a finite set of directions, including for periodic sequences.

Findings

Sequence of Steiner symmetrizations converges.

Limit body is symmetric under directions used infinitely often.

Periodic sequences of symmetrizations always converge.

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.