A Geometric Picture of Entanglement and Bell Inequalities

We work in the real Hilbert space $\Ha_s$ of hermitian Hilbert-Schmid operators and show that the entanglement witness which shows the maximal violation of a generalized Bell inequality (GBI) is a tangent functional to the convex set $S \subset \Ha_s$ of separable states. This violation equals the euclidean distance in $\Ha_s$ of the entangled state to $S$ and thus entanglement, GBI and tangent functional are only different aspects of the same geometric picture. This is explicitly illustrated in the example of two spins, where also a comparison with familiar Bell inequalities is presented.