Steiner symmetrization on the sphere

TL;DR

We extend Steiner symmetrization to the sphere by defining a spherical symmetrization $\sigma_{L,H}$ along distance curves with axis $L$ inside an open hemisphere $S\subset \mathbb{S}^n$. We prove volume is preserved for sets with the connectedness property, and establish convexity preservation on $\mathbb{S}^2$ under an angular monotonicity condition, while convexity can fail for $n>2$. The paper further shows nonincreasing spherical perimeter and diameter in the planar case, and demonstrates that repeated symmetrizations can approximate a spherical cap of the same area, enabling spherical analogues of classic Euclidean results. Applications include a Sas-type extremal theorem and a floating-area isoperimetric inequality for centrally symmetric disks, as well as a spherical Winternitz-type theorem grounded in spherical centroids and moments. Overall, the work provides a systematic spherical counterpart to Euclidean Steiner symmetrization, with implications for isoperimetric and approximation problems on $\mathbb{S}^n$.

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.

Figures (3)

• Figure 1: An illustration for the dissection of $P$ in the proof of Theorem \ref{['thm:planarconvexity']}. The continuous line in the figure indicates $\mathop{\mathrm{bd}}\nolimits (P)$. The dashed segment in $P$, which is a rotated copy of the left segment in $\mathop{\mathrm{bd}}\nolimits(P)$ in the figure around the poles, decomposes $P$ into the regions $P_1, P_2$. The symbols $\varphi_2, \varphi_1-\varphi_2$ show the angular lengths of the corresponding distance curve arcs in the boundaries of $P_1$ and $P_2$.
• Figure 2: An illustration for the proof of Lemma \ref{['lem:ellipse']}.
• Figure 3: An illustration for the proof of Theorem \ref{['spherical-macbeath-thm']} with $N=6$. The dotted arcs denote the chords of $K$ on the distance curves $C_i$. The dashed polygonal curves denote the boundaries of the spherical inscribed polygons $P$ and $Q$.

Theorems & Definitions (60)

• Definition 2.1
• Remark 2.2
• Remark 2.3
• Remark 2.4
• Remark 2.5
• Definition 2.6
• Theorem 2.7
• proof
• Remark 2.8
• Definition 2.9