  Entangled many-body states are an essential resource for quantum computing
and interferometry. Determining the type of entanglement present in a system
usually requires access to an exponential number of parameters. We show that in
the case of pure multi-particle quantum states, features of the global
entanglement can already be extracted from local information alone. This is
achieved by associating with any given class of entanglement an entanglement
polytope---a geometric object which characterizes the single-particle states
compatible with that class. Our results, applicable to systems of arbitrary
size and statistics, give rise to local witnesses for global pure-state
entanglement, and can be generalized to states affected by low levels of noise.
