We prove that a smooth convex body of diameter δ<π2\documentclass[12pt]{minimal}
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\begin{document}$$\delta < \frac{\pi }{2}$$\end{document} on the d-dimensional unit sphere Sd\documentclass[12pt]{minimal}
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\begin{document}$$S^d$$\end{document} is of constant diameter δ\documentclass[12pt]{minimal}
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\begin{document}$$\delta $$\end{document} if and only if it is of constant width δ\documentclass[12pt]{minimal}
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\begin{document}$$\delta $$\end{document}. We also show this equivalence for all convex bodies on S2\documentclass[12pt]{minimal}
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\begin{document}$$S^2$$\end{document}. Since, as shown earlier, the equivalence on Sd\documentclass[12pt]{minimal}
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\begin{document}$$S^d$$\end{document} is true for every δ≥π2\documentclass[12pt]{minimal}
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\begin{document}$$\delta \ge \frac{\pi }{2}$$\end{document}, the question whether spherical bodies of constant diameter and constant width on Sd\documentclass[12pt]{minimal}
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\begin{document}$$S^d$$\end{document} coincide remains open for non-smooth bodies on Sd\documentclass[12pt]{minimal}
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\begin{document}$$S^d$$\end{document}, where d≥3\documentclass[12pt]{minimal}
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\begin{document}$$d\ge 3$$\end{document}.
