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

{\bf Abstract.} Let $D$ be a convex body of diameter $\delta$, where $0 < \delta < \frac{\pi}{2}$, on the $d$-dimensional sphere. We prove that $D$ is of constant diameter $\delta$ if and only if it is of constant width $\delta$ in the following two cases. The first case is when $D$ is smooth. The second case is when $d=2$.