Convergence properties of symmetrization processes

TL;DR

This paper studies a family of symmetrization processes, including Steiner symmetrization, demonstrating their convergence properties and identifying common behaviors and pathological phenomena among them.

Contribution

It introduces a broad class of symmetrizations with shared convergence behaviors and analyzes their properties and potential irregularities.

Findings

All symmetrizations in the family share the same convergence process.

The paper identifies pathological phenomena in these symmetrizations.

Steiner symmetrization is a special case within this family.

Abstract

Steiner symmetrization is well known for its rounding and general convergence properties. We identify a whole family of symmetrizations sharing analogue behaviors: In fact we prove that all these symmetrizations share the same converging symmetrization processes, together with some pathological phenomena.