Symmetrization in Geometry

TL;DR

This paper introduces a unified framework for various symmetrization processes in convex geometry, explores their properties, and establishes dual relationships and characterizations for classical and new symmetrizations.

Contribution

It develops the concept of $i$-symmetrization, provides new formulas for Steiner and Minkowski symmetrals, and characterizes key symmetrizations through natural properties.

Findings

Dual relationship between Steiner and Minkowski symmetrals

Characterizations of classical symmetrizations based on natural properties

Introduction of new symmetrization processes

Abstract

The concept of an $i$-symmetrization is introduced, which provides a convenient framework for most of the familiar symmetrization processes on convex sets. Various properties of $i$-symmetrizations are introduced and the relations between them investigated. New expressions are provided for the Steiner and Minkowski symmetrals of a compact convex set which exhibit a dual relationship between them. Characterizations of Steiner, Minkowski and central symmetrization, in terms of natural properties that they enjoy, are given and examples are provided to show that none of the assumptions made can be dropped or significantly weakened. Other familiar symmetrizations, such as Schwarz symmetrization, are discussed and several new ones introduced.