A Note on the Carathéodory Number of the Joint Numerical Range

We show that the Carathéodory number of the joint numerical range of d many bounded self-adjoint operators is at most d-1, and even at most d-2 if the underlying Hilbert space has dimension at least 3. This extension of the classical convexity results for numerical ranges shows that also joint numerical ranges are significantly less non-convex than general sets.