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Rearrangement and polarization

By Gabriele Bianchi, Richard J. Gardner, Paolo Gronchi and Markus Kiderlen
CSGB Research Reports
No. 12, December 2019

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 $\mathbb{R}^n$. 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.

Keywords: Convex body, Steiner symmetrization, Schwarz symmetrization, Minkowski symmetrization, rearrangement, polarization

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