Disk matrices and the proximal mapping for the numerical radius

Optimal matrices for problems involving the matrix numerical radius often have fields of values that are disks, a phenomenon associated with partial smoothness. Such matrices are highly structured: we experiment in particular with the proximal mapping for the radius, which often maps n-by-n random matrix inputs into a particular manifold of disk matrices that has real codimension 2n. The outputs, computed via semidefinite programming, also satisfy an unusual rank property at optimality.