  Solutions to optimization problems involving the numerical radius often
belong to a special class: the set of matrices having field of values a disk
centered at the origin. After illustrating this phenomenon with some examples,
we illuminate it by studying matrices around which this set of "disk matrices"
is a manifold with respect to which the numerical radius is partly smooth. We
then apply our results to matrices whose nonzeros consist of a single
superdiagonal, such as Jordan blocks and the Crabb matrix related to a
well-known conjecture of Crouzeix. Finally, we consider arbitrary complex
three-by-three matrices; even in this case, the details are surprisingly
intricate. One of our results is that in this real vector space with dimension
18, the set of disk matrices is a semi-algebraic manifold with dimension 12.

Numerical Analysis;Optimization and Control