Generalization of Klain's Theorem to Minkowski Symmetrization of compact sets and related topics

TL;DR

This paper extends Klain's theorem to Minkowski symmetrization of compact sets, proving convergence results and exploring symmetrization properties, including idempotency and approximation of balls, with implications for geometric analysis.

Contribution

It generalizes Klain's theorem to Minkowski symmetrization of compact sets and investigates symmetrization properties like idempotency and approximation of spheres.

Findings

Proved convergence of Minkowski symmetrals for compact sets.

Established idempotency conditions for specific symmetrization families.

Extended approximation results of balls via finite symmetrizations.

Abstract

We shall prove a convergence result relative to sequences of Minkowski symmetrals of general compact sets. In particular, we investigate the case when this process is induced by sequences of subspaces whose elements belong to a finite family, following the path marked by Klain in [13], and the generalizations in [4] and [2]. We prove an analogue result for Fiber symmetrization of a specific class of compact sets. The idempotency for symmetrization of this family of sets is investigated, leading to a simple generalization of a result from Klartag [14] regarding the approximation of a ball through a finite number of symmetrizations, and generalizing an approximation result in [9]