Let $A_1,A_2,\dots,A_M$ be given $N\times N$ hermitian matrices. The numerical range is defined as the set \begin{align} \mathbb{S}=\{(u^HA_1u,\dots,u^HA_Mu)\in \mathbb{R}^M\mid u^Hu=1\} \end{align} By toeplitz hausdorff theorem, $\mathbb{S}$ is convex for

- $M=2$, $N\geq 2$, no conditions on $A_i$
- $M=3$, $N\geq 3$, no conditions on $A_i$
- $A_i$'s are tridiagonal and real in some basis. No conditions on $M$ and $N$. (from Nathaniel Johnston's answer).

I would like to know the other cases where $\mathbb{S}$ is convex. Can some body point me to known references?