Convexity of the joint numerical range: topological and differential geometric viewpoints

We investigate the convexity of the joint numerical range of m-tuples of n×n hermitian matrices. The methods come from differential geometry and the differential and algebraic topology. Our main result is a sufficient condition for convexity of the joint numerical range for arbitrary m and n. Modulo a mild technical assumption, this condition is formulated in terms of the largest eigenvalue of an associated family of hermitian matrices parameterized by the (m−1)-dimensional unit sphere. The condition is that the eigenvalue has constant multiplicity. We show that the constant multiplicity condition is in fact a criterion for the stable convexity of numerical ranges. As a byproduct of our main result, we obtain a new proof of the celebrated Toeplitz–Hausdorff theorem, and of its extension: The numerical range of any triple of hermitian n×n matrices is convex if n>2.