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Rearrangement and polarization

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  • Gabriele Bianchi, University of Florence
  • ,
  • Richard J. Gardner, Western Washington University
  • ,
  • Paolo Gronchi, University of Florence
  • ,
  • Markus Kiderlen

The paper has two main goals. The first is to take a new approach to rearrangements on certain classes of measurable real-valued functions on Rn. Rearrangements are maps that are monotonic (up to sets of measure zero) and equimeasurable, i.e., they preserve the measure of super-level sets of functions. All the principal known symmetrization processes for functions, such as Steiner and Schwarz symmetrization, are rearrangements, and these have a multitude of applications in diverse areas of the mathematical sciences. The second goal is to understand which properties of rearrangements characterize polarization, a special rearrangement that has proved particularly useful in a number of contexts. In order to achieve this, new results are obtained on the structure of measure-preserving maps on convex bodies and of rearrangements generally.

Original languageEnglish
Article number107380
JournalAdvances in Mathematics
Publication statusPublished - Nov 2020

    Research areas

  • Minkowski symmetrization, Polarization, Rearrangement, Schwarz symmetrization, Steiner symmetrization, Two-point symmetrization

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