Symmetrization in Geometry

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.

arXiv commentA chacterization of central symmetrization has been added and several typos have been corrected. This version has been accepted for publication on Advances in Mathematics

References in corpus (2)

• A characterization of Blaschke addition
• The role of the Rogers-Shephard inequality in the characterization of the difference body

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• Affine Isoperimetric Inequalities for Higher-Order Projection and Centroid Bodies
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• The Pólya-Szegő inequality for smoothing rearrangements
• Iterated-logarithm laws for convex hulls of random walks with drift
• Some extremal problems for polygons in the Euclidean plane
• Minkowski additive operators under volume constraints
• An affine Orlicz Polya-Szego principle
• The Petty projection inequality for sets of finite perimeter
• Affine Orlicz Pólya-Szegö principles and their equality cases
• Generalization of Klain's Theorem to Minkowski Symmetrization of compact sets and related topics
• Volumetric bounds for intersections of congruent balls