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 

 1. $M=2$, $N\geq 2$, no conditions on $A_i$
 2. $M=3$, $N\geq 3$, no conditions on $A_i$
 3. $A_i$'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?