The intersection L of two different non-opposite hemispheres G and H of the ddimensional unit 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 convex body C ⊂ S d we define width G (C) as the thickness of the narrowest lune or lunes of the form G ∩ H containing C. If width 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 < π 2 , then C is strictly convex. Moreover, we check when spherical bodies of constant width and constant diameter coincide. Mathematics Subject Classification. 52A55.A spherical ball B ρ (x) of radius ρ ∈ (0, π 2 ], or a ball for short, is the set of points of S d at distances at most ρ from a fixed point x, which is called the center of this ball. An open ball (a sphere) is the set of points of S d having distance smaller than (respectively, exactly) ρ from a fixed point. A spherical ball of radius π 2 is called a hemisphere. So it is the common part of S d and a closed half-space of E d+1 . We denote by H(m) the hemisphere with center m. Two hemispheres with centers at a pair of antipodes are called opposite.A spherical (d − 1)-dimensional ball of radius ρ ∈ (0, π 2 ] is the set of points of a (d − 1)-dimensional great sphere of S d which are at distances at most ρ from a fixed point. We call it the center of this ball. The (d − 1)-dimensional balls of radius π 2 are called (d − 1)-dimensional hemispheres, and semicircles for d = 2.A set C ⊂ S d is said to be convex if no pair of antipodes belongs to C and if for every a, b ∈ C we have ab ⊂ C. A closed convex set on S d with nonempty interior is called a convex body. Some basic references on convex bodies and their properties are [4], [9] and [10]. A short survey of other definitions of convexity on S d is given in Section 9.1 of [2].Since the intersection of every family of convex sets is also convex, for every set A ⊂ S d contained in an open hemisphere of S d there is the smallest convex set conv(A) containing Q. We call it the convex hull of A.Let C ⊂ S d be a convex body. Let Q ⊂ S d be a convex body or a hemisphere. We say that C touches Q from inside if C ⊂ Q and bd(C)∩bd(Q) = ∅. We say that C touches Q from outside if C ∩ Q = ∅ and int(C) ∩ int(Q) = ∅. In both cases, points of bd(C) ∩ bd(Q) are called points of touching. In the first case, if Q is a hemisphere, we also say that Q supports C, or supports C at t, provided t is a point of touching. If at every boundary point of C exactly one hemisphere supports C, we say that C is smooth. We call e ∈ C an extreme point of C if C \ {e} is convex.If hemispheres G and H of S d are different and not opposite, then L = G∩H is called a lune of S d . This notion is considered in many books and papers (for instance, see [12]).

The intersection L of two different non-opposite hemispheres of the unit sphere S 2 is called a lune. By ∆(L) we denote the distance of the centers of the semicircles bounding L. By the thickness ∆(C) of a convex body C ⊂ S 2 we mean the minimal value of ∆(L) over all lunes L ⊃ C. We call a convex body R ⊂ S 2 reduced provided ∆(Z) < ∆(R) for every convex body Z being a proper subset of R. Our aim is to estimate the diameter of R, where ∆(R) < π 2 , in terms of its thickness.2010 Mathematics Subject Classification. 52A55.