Geometric picture of entanglement and Bell inequalities

We work in the real Hilbert space ${\mathcal{H}}_s$ of Hermitian Hilbert-Schmidt 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 $S\subset {\mathcal{H}}_s$ of separable states. This violation equals the Euclidean distance in ${\mathcal{H}}_s$ 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.