Spherical quadrilateral with three right angles and its application for diameter of extreme points of a convex body

Abstract

We prove a theorem on the relationships between the lengths of sides of a spherical quadrilateral with three right angles. They are analogous to the relationships in the Lambert quadrilateral in the hyperbolic plane. We apply this theorem in the proof of our second theorem that if $C$ is a two-dimensional spherical convex body of diameter $\delta \in (\frac{1}{2}\pi,\pi)$, then the diameter of the set of extreme points of $C$ is at least $2 \arccos \big(\frac{1}{4}(\cos \delta + \sqrt {\cos^2 \delta +8})\big)$. This estimate cannot be improved.