Spherical bodies of constant width

The intersection $L$ of two different non-opposite hemispheres $G$ and $H$ of a $d$-dimensional sphere $S^d$ is called a lune. By the thickness of $L$ we mean the distance of the centers of the $(d-1)$-dimensional hemispheres bounding $L$. For a hemisphere $G$ supporting a %spherical convex body $C \subset S^d$wedefine${\rm width}_G©$ as the thickness of the narrowest lune or lunes of the form $G\cap H$ containing $C$. If ${\mathrm{w}\mathrm{i}\mathrm{d}\mathrm{t}\mathrm{h}}_G(C)=w$ for every hemisphere $G$ supporting $C$, we say that $C$ is a body of constant width $w$. We present properties of these bodies. In particular, we prove that the diameter of any spherical body $C$ of constant width $w$ on $S^d$ is $w$, and that if $w<\frac{\pi }{2}$, then $C$ is strictly convex. Moreover, we are checking when spherical bodies of constant width and constant diameter coincide.