Journal of Convex Analysis 33 (2026), No. 1&2, 517--519
Copyright Heldermann Verlag 2026
A Note on the Carathéodory Number of the Joint Numerical Range
Beatrice Maier
Department of Mathematics, University of Innsbruck, Austria
Beatrice.Maier@student.uibk.ac.at
Tim Netzer
Department of Mathematics, University of Innsbruck, Austria
Tim.Netzer@uibk.ac.at
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.
Keywords: Joint numerical range, convexity, Caratheodory number.
MSC: 47A12, 52A05.
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