When is a spherical body of constant diameter of constant width?

We prove that a smooth convex body of diameter δ < π 2 on the d-dimensional unit sphere S d is of constant diameter δ if and only if it is of constant width δ . We also show this equivalence for all convex bodies on S 2 . Since, as shown earlier, the equivalence on S d is true for every δ ≥ π 2 , the question whether spherical bodies of constant diameter and constant width on S d coincide remains open for non-smooth bodies on S d , where d ≥ 3 . [ABSTRACT FROM AUTHOR]