A Note on the Carath\'eodory Number of the Joint Numerical Range
Abstract
We show that the Carath\'eodory 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.
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