Spherical Steiner Symmetrizations

Abstract

In this paper, we primarily investigate and establish several properties of spherical Steiner symmetrizations, along with the isoperimetric property of the spherical cap in Sn. Specifically, we study the monotonically decreasing property of the measure of the symmetric difference of two spherical compact sets, the monotonically decreasing property of the spherical diameter of a spherical compact set, the convergence of iterative spherical Steiner symmetrizations, and so on. In particular, we prove that the sequence of iterative spherical Steiner symmetrizations of K⊂Sn, which follow sequences selected from a finite set of directions, converges to a spherical cap with the same measure as K, extending the result from Rn to Sn on Steiner symmetrizations. It provides us with valuable insights for studying the relevant applications and conclusions of spherical Steiner symmetrizations.