  We computationally investigate the complete polytope of Bell inequalities for
2 particles with small numbers of possible measurements and outcomes. Our
approach is limited by Pitowsky's connection of this problem to the
computationally hard NP problem. Despite this, we find that there are very few
relevant inequivalent inequalities for small numbers. For example, in the case
with 3 possible 2-outcome measurements on each particle, there is just one new
inequality. We describe mixed 2-qubit states which violate this inequality but
not the CHSH. The new inequality also illustrates a sharing of bi-partite
non-locality between three qubits: something not seen using the CHSH
inequality. It also inspires us to discover a class of Bell inequalities with m
possible n-outcome measurements on each particle.
