On reduced spherical bodies

TL;DR

This thesis explores properties of reduced spherical convex bodies, especially those of constant width, providing new insights into their shape, diameter, covering disks, and containment properties on spheres.

Contribution

It offers new characterizations and bounds for reduced bodies of constant width and diameter on spheres, advancing understanding of spherical convex geometry.

Findings

Reduced bodies of thickness less than π/2 have specific shape properties.

Bodies of constant width on S^d exhibit particular geometric relations.

Every reduced spherical polygon is contained in a disk of radius equal to its thickness.

Abstract

This thesis consists of five papers about reduced spherical convex bodies and in particular spherical bodies of constant width on the $d$-dimensional sphere $S^d$. In paper I we present some facts describing the shape of reduced bodies of thickness under $\frac{\pi}{2}$ on $S^2$. We also consider reduced bodies of thickness at least $\frac{\pi}{2}$, which appear to be of constant width. Paper II focuses on bodies of constant width on $S^d$. We present the properties of these bodies and in particular we discuss conections between notions of constant width and of constant diameter. In paper III we estimate the diameter of a reduced convex body. The main theme of paper IV is estimating the radius of the smallest disk that covers a reduced convex body on $S^2$. The result of paper V is showing that every spherical reduced polygon $V$ is contained in a disk of radius equal to the thickness of this body centered at a boundary point of $V$.