  Similarly to the classic notion in $E^d$, a subset of a positive diameter
below $\frac{\pi}{2}$ of a hemisphere of the sphere $S^d$ is called complete,
provided adding any extra point increases its diameter. Complete sets are
convex bodies on $S^d$. Our main theorem says that on $S^d$ complete bodies of
diameter $\delta$ coincide with bodies of constant width $\delta$.
