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? 



Known Results on Convexity of Numerical Range
The only result of this nature that I'm aware of (beyond the two cases involving $M$ and $N$ that you've listed) is Theorem 3.1 in "C.-K. Li and Y.-T. Poon. [Convexity of the joint numerical range](http://people.wm.edu/~cklixx/poon-1.pdf). SIAM J. Matrix Analysis Appl. 21 (1999), 668-678", which basically says that the numerical range is convex when each of the $A\_i$'s are tridiagonal and real in some basis.


