Let $A \in M_n(\mathbb{C})$. The numerical range of $A$ is denoted as $$W(A) = \left \{ \frac{x^*Ax}{x^*x} \mid x \in \mathbb{C}^n, x \ne 0 \right \}.$$
Is there anything known about the boundary of $A$ ? Can we relate the vectors $x$ that achieve the boundary values of $W(A)$ to some properties about $A$ ?
For example, if $A$ is normal, then the boundary of $W(A)$ is just the convex hull of the eigenvalues of $A$ (see here). Can we say something concerte in the general case ?