Symmetrization in geometry

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• 作者：Gabriele Bianchi^a ; ^{gabriele.bianchi@unifi.it} ; Richard J. Gardner^b ; ^{richard.gardner@wwu.edu} ; Paolo Gronchi^c ; ^{paolo.gronchi@unifi.it}
• 关键词：primary ; 52A20 ; 52A39 ; secondary ; 28B20 ; 52A38 ; 52A40
• 刊名：Advances in Mathematics
• 出版年：2017
• 出版时间：14 January 2017
• 年：2017
• 卷：306
• 期：Complete
• 页码：51-88
• 全文大小：678 K
• 卷排序：306

文摘

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.