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Since spherical buildings are CAT(1), we get a fixed point if $\mathop{\rm diam}S<\tfrac \pi 2$.$$\mathop{\rm rad}S<\tfrac \pi 2.$$
Since spherical buildings are CAT(1), we get a fixed point if $\mathop{\rm diam}S<\tfrac \pi 2$.
Since spherical buildings are CAT(1), we get a fixed point if $$\mathop{\rm rad}S<\tfrac \pi 2.$$
