Spherical bodies of constant width

The intersection L of two different non-opposite hemispheres G and H of the d-dimensional unit sphere is called a lune. By the thickness of L we mean the distance of the centers of the -dimensional hemispheres bounding L. For a hemisphere G supporting a convex body we define as the thickness of the narrowest lune or lunes of the form containing C. If 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 is w, and that if , then C is strictly convex. Moreover, we check when spherical bodies of constant width and constant diameter coincide.