The area of reduced spherical polygons

TL;DR

This paper proves two conjectures regarding the area of reduced spherical polygons, showing that regular polygons maximize area among reduced spherical polygons of the same thickness, with implications for understanding spherical geometry.

Contribution

It confirms Lassak's conjectures, establishing that regular spherical polygons have maximal area among reduced spherical polygons of the same thickness.

Findings

Reduced spherical non-regular n-gons have smaller area than regular ones of same thickness.

The area of any reduced spherical polygon is less than that of regular odd-gons with the same thickness and increasing vertices.

The results validate conjectures about maximal area properties of regular spherical polygons.

Abstract

We confirm two conjectures of Lassak on the area of reduced spherical polygons. The area of every reduced spherical non-regular $n$-gon is less than that of the regular spherical $n$-gon of the same thickness. Moreover, the area of every reduced spherical polygon is less than that of the regular spherical odd-gons of the same thickness and whose number of vertices tends to infinity.