  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 Peyerimhoff (J. London
Math. Soc. (2) 66: 753-768, 2002). 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. We also prove a spherical variant of a theorem of
Winternitz.
