Steiner symmetrization on the sphere

TL;DR

This paper introduces a spherical version of Steiner symmetrization, demonstrating its properties, convergence behavior, and applications to spherical geometry, including analogues of classical theorems and conjectures.

Contribution

It generalizes Steiner symmetrization to spherical space, analyzes its properties, and applies it to prove new geometric results and conjectures in spherical geometry.

Findings

Preserves volume in all dimensions

Maintains convexity in the spherical plane but not in higher dimensions

Converges to spherical caps under certain conditions

Abstract

The aim of this paper is to introduce a generalization of Steiner symmetrization in Euclidean space for spherical space, which is the dual of the Steiner symmetrization in hyperbolic space introduced by J. Schneider (Manuscripta Math. 60: 437-461, 1988). We show that this symmetrization preserves volume in every dimension, and convexity in the spherical plane, but not in dimensions $n > 2$. In addition, we investigate the monotonicity properties of the perimeter and diameter of a set under this process, and find conditions under which the image of a spherically convex disk under a suitable sequence of Steiner symmetrizations converges to a spherical cap. We apply our results to prove a spherical analogue of a theorem of Sas, and to confirm a conjecture of Besau and Werner (Adv. Math. 301: 867-901, 2016) for centrally symmetric spherically convex disks. Lastly, we prove a spherical variant of a theorem of Winternitz.