= Minimum enclosing ball in hyperbolic space
For a nonempty finite set $\{p_1,\ldots,p_m\}$ in finite-dimensional <hyperbolic space>, the continuous function $R(x)=\max_jd(x,p_j)$ attains its minimum. Indeed, a bounded candidate radius confines all possible improving centers to a closed bounded ball about $p_1$. Such a ball is compact in the <Poincare ball model>, since its points remain away from the ideal boundary. Minimizing $R$ on it supplies a <closed ball> of smallest enclosing radius. More generally this argument works for any nonempty bounded set in a <proper metric space> when the supremum-distance function is finite.
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