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

Abstract

We shall prove a convergence result relative to sequences of Minkowski symmetrals of 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 [11], and the generalizations in [4] and [2].We prove an analogue result for Fiber symmetrization of a specific class of compact sets, namely the convex shells. The idempotency degree for symmetrization of this family of sets is investigated, leading to a simple generalization of a result from Klartag [12] regarding the approximation of a ball through a finite number of symmetrizations.Two counterexamples to convergence of sequences of symmetrals in the plane are proven, extending some ideas in [2] to a wider class of i-symmetrizations, which include the Minkowski one.