  Black-hole spacetimes with a "photonsphere", a hypersurface on which massless
particles can orbit the black hole on circular null geodesics, are studied. We
prove that among all possible trajectories (both geodesic and non-geodesic)
which circle the central black hole, the null circular geodesic is
characterized by the {\it shortest} possible orbital period as measured by
asymptotic observers. Thus, null circular geodesics provide the fastest way to
circle black holes. In addition, we conjecture the existence of a universal
lower bound for orbital periods around compact objects (as measured by
flat-space asymptotic observers): $T_{\infty}\geq 4\pi M$, where $M$ is the
mass of the central object. This bound is saturated by the null circular
geodesic of the maximally rotating Kerr black hole.
