LAUSR

We prove that a smooth convex body of diameter $$\delta < \frac{\pi }{2}$$ δ < π 2 on the d -dimensional unit sphere $$S^d$$ S d is of constant diameter $$\delta $$ δ if and only if it is of constant width $$\delta $$ δ . We also show this equivalence for all convex bodies on $$S^2$$ S 2 . Since, as shown earlier, the equivalence on $$S^d$$ S d is true for every $$\delta \ge \frac{\pi }{2}$$ δ ≥ π 2 , the question whether spherical bodies of constant diameter and constant width on $$S^d$$ S d coincide remains open for non-smooth bodies on $$S^d$$ S d , where $$d\ge 3$$ d ≥ 3 .