# Convergence of symmetrization processes

## TL;DR

This paper investigates the convergence properties of various symmetrization processes in Euclidean space, establishing new universal sequences of subspaces that lead to symmetric limits for a broad class of sets.

## Contribution

It introduces new universal sequences for symmetrizations, extending Klain's theorem to multiple symmetrization types, and characterizes conditions for convergence to symmetric sets.

## Key findings

- New universal sequences for symmetrizations are identified.
- Klain's theorem is extended to Schwarz, Minkowski, Minkowski-Blaschke, and fiber symmetrizations.
- Convergence to symmetric sets is characterized for various symmetrization processes.

## Abstract

Steiner and Schwarz symmetrizations, and their most important relatives, the Minkowski, Minkowski-Blaschke, fiber, inner rotational, and outer rotational symmetrizations, are investigated. The focus is on the convergence of successive symmetrals with respect to a sequence of $i$-dimensional subspaces of $\mathbb{R}^n$. Such a sequence is called universal for a family of sets if the successive symmetrals of any set in the family converge to a ball with center at the origin. New universal sequences for the main symmetrizations, for all valid dimensions $i$ of the subspaces, are found, by combining two groups of results. The first, published separately, provides finite sets ${\mathcal{F}}$ of subspaces such that reflection symmetry (or rotational symmetry) with respect to each subspace in ${\mathcal{F}}$ implies full rotational symmetry. In the second, proved here, a theorem of Klain for Steiner symmetrization is extended to Schwarz, Minkowski, Minkowski-Blaschke, and fiber symmetrizations, showing that if a sequence of subspaces is drawn from a finite set ${\mathcal{F}}$ of subspaces, the successive symmetrals of any compact convex set converge to a compact convex set that is symmetric with respect to any subspace in ${\mathcal{F}}$ appearing infinitely often in the sequence. It is also proved that for Steiner, Schwarz, and Minkowski symmetrizations, a sequence of $i$-dimensional subspaces is universal for the class of compact sets if and only if it is universal for the class of compact convex sets, and Klain's theorem is shown to hold for Schwarz symmetrization of compact sets.