Symmetrization in geometry
Copyright policy )Symmetrization in geometry
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.
A chacterization of central symmetrization has been added and several typos have been corrected. This version has been accepted for publication on Advances in Mathematics
- University of Florence Italy
- Eastern Washington University United States
- New York University Italy
Schwarz symmetrization, Minkowski symmetrization, Primary: 52A20, 52A39, secondary: 28B20, 52A38, 52A40, Metric Geometry (math.MG), Steiner symmetrization, Convex sets in \(n\) dimensions (including convex hypersurfaces), Mixed volumes and related topics in convex geometry, Length, area, volume and convex sets (aspects of convex geometry), convex body, Mathematics - Metric Geometry, Simmetrizzazione, Simmetrizzazione di Steiner, Simmetrizzazione di Minkowski, central symmetrization, Inequalities and extremum problems involving convexity in convex geometry, FOS: Mathematics, Set-valued set functions and measures; integration of set-valued functions; measurable selections
Schwarz symmetrization, Minkowski symmetrization, Primary: 52A20, 52A39, secondary: 28B20, 52A38, 52A40, Metric Geometry (math.MG), Steiner symmetrization, Convex sets in \(n\) dimensions (including convex hypersurfaces), Mixed volumes and related topics in convex geometry, Length, area, volume and convex sets (aspects of convex geometry), convex body, Mathematics - Metric Geometry, Simmetrizzazione, Simmetrizzazione di Steiner, Simmetrizzazione di Minkowski, central symmetrization, Inequalities and extremum problems involving convexity in convex geometry, FOS: Mathematics, Set-valued set functions and measures; integration of set-valued functions; measurable selections
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