Covering a reduced spherical body by a disk

TL;DR

This paper establishes new theorems on covering spherical convex bodies with disks, providing explicit bounds based on their width and thickness, advancing geometric understanding of spherical coverings.

Contribution

It proves two novel theorems giving explicit disk covering bounds for spherical convex bodies with specific width and thickness constraints.

Findings

Convex bodies of width ≥ π/2 can be covered by a disk with a specific radius.

Reduced convex bodies of thickness < π/2 can be covered by a disk with a different explicit radius.

Provides formulas for disk radii based on geometric parameters.

Abstract

In this paper, the following two theorems are proved: $(1)$ every spherical convex body $W$ of constant width $\Delta (W) \geq \frac{\pi}{2}$ may be covered by a disk of radius $\Delta(W) + \arcsin \left( \frac{2\sqrt{3}}{3} \cdot \cos \frac{\Delta(W)}{2}\right) - \frac{\pi}{2}$; $(2)$ every reduced spherical convex body $R$ of thickness $\Delta(R)<\frac{\pi}{2}$ may be covered by a disk of radius $\arctan \left( \sqrt{2} \cdot \tan \frac{\Delta(R)}{2}\right)$.