  We present a method to calculate, without making assumptions about the local
dark matter velocity distribution, the maximal and minimal number of signal
events in a direct detection experiment given a set of constraints from other
direct detection experiments and/or neutrino telescopes. The method also allows
to determine the velocity distribution that optimizes the signal rates. We
illustrate our method with three concrete applications: i) to derive a
halo-independent upper limit on the cross section from a set of null results,
ii) to confront in a halo-independent way a detection claim to a set of null
results and iii) to assess, in a halo-independent manner, the prospects for
detection in a future experiment given a set of current null results.
