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

TL;DR

This paper establishes a relationship between side lengths of a specific spherical quadrilateral and applies it to determine a lower bound on the diameter of extreme points in spherical convex bodies, with a proof of optimality.

Contribution

It introduces a new theorem on spherical quadrilaterals with three right angles and applies it to convex geometry to derive a sharp lower bound on the diameter of extreme points.

Findings

Proved a theorem relating side lengths of a spherical quadrilateral with three right angles.

Derived a lower bound for the diameter of extreme points in spherical convex bodies.

Showed that the estimate for the diameter cannot be improved.

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.