On some problems concerning symmetrization operators

TL;DR

This paper investigates open questions about symmetrization operators on convex sets, building on prior systematic studies to deepen understanding of their properties and applications in geometry.

Contribution

It addresses specific open questions posed in previous work, advancing the theoretical understanding of symmetrization operators in convex geometry.

Findings

Resolved several open questions from prior research

Enhanced understanding of symmetrization operator properties

Contributed to the theoretical framework of convex set symmetrization

Abstract

In [G. Bianchi, R. J. Gardner and P. Gronchi, Symmetrization in Geometry, Adv. Math., vol. 306 (2017), 51-88], a systematic study of symmetrization operators on convex sets and their properties is conducted. In the end of their article, the authors pose several open questions. The primary goal of this manuscript is to study these questions.