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Convexity of the Joint Numerical Range: Topological and Differential Geometric Viewpoints
Eugene Gutkin, Edmond Jonckheere and Michael Karow
We show that the outer boundary of the joint numerical range of any number, $m$, of hermitian matrices is convex if the multiplicity of the largest eigenvalue of the associated hermitian matrix is constant. Thus for $m>3$ the problem of convexity of the joint numerical range is essentially topological.
Furthermore, our sufficient condition for convexity of the joint numerical range is also necessary for its stable convexity.