The Perimeter and Area of Reduced Spherical Polygons of Thickness $$\pi /2$$

Abstract

We confirm a few recent conjectures of Lassak on the perimeter and area of reduced spherical polygons of thickness $$\pi /2$$ . This paper is based on the study of the sufficient and necessary conditions whether a spherical polygon of thickness $$\pi /2$$ is reduced. The perimeter (resp. area) of every reduced spherical k-gon of thickness $$\pi /2$$ is not greater than that of the regular spherical triangle (resp. regular spherical n-gon) of thickness $$\pi /2$$ , where $$3\le k\le n$$ . Moreover, the regular spherical odd-gon with n vertices and thickness $$\pi /2$$ has the minimum perimeter (resp. maximum area) among all reduced spherical k-gons of thickness $$\pi /2$$ , where $$3\le k\le n$$ .