  Bell inequalities are derived for any number of observers, any number of
alternative setups for each one of them, and any number of distinct outcomes
for each experiment. It is shown that if a physical system consists of several
distant subsystems, and if the results of tests performed on the latter are
determined by local variables with objective values, then the joint
probabilities for triggering any given set of distant detectors are convex
combinations of a finite number of Boolean arrays, whose components are either
0 or 1 according to a simple rule. This convexity property is both necessary
and sufficient for the existence of local objective variables. It leads to a
simple graphical method which produces a large number of generalized Clauser-
Horne inequalities corresponding to the faces of a convex polytope. It is
plausible that quantum systems whose density matrix has a positive partial
transposition satisfy all these inequalities, and therefore are compatible with
local objective variables, even if their quantum properties are essentially
nonlocal.
