Is $W(S_1,S_2)$ convex? Let $F=\mathbb{C}^4$  be endowed with the norm $\|\cdot\|_2$. Let the operators $$
S_1=\left(\begin{array}{cccc}0&1&0&0\\0&0&0&0\\0&0&0&0\\0&0&0&0\end{array}\right)\;\mbox{and}\;\;S_2=\left(\begin{array}{cccc}0&0&0&0\\0&0&0&0\\0&0&1&0\\0&0&0&1\end{array}\right)
.$$
The numerical range of $(S_1,S_2)$ is given by
$$W(S_1,S_2) =\left\{\left(\overline{a}b,|c|^2+|d|^2\right);\;(a,b,c,d) \in \mathbb{C}^4\;\;\hbox{and}\;|a|^2+|b|^2+|c|^2+|d|^2=1\right\}.$$

Is $W(S_1,S_2)$ convex?

We can see that $(0,1),\,(\frac{1}{2},0)\in W(S_1,S_2)$. I hope to find a point in the segment joining $(0,1)$ and $(\frac{1}{2},0)$ which is not in $W(S_1,S_2)$.

The open line segment is $\{(t/2,1-t)|0<t<1\}.$  Given $0<t<1,$ choose any $c,d$ such that $|c|^2+|d|^2=1-t$ for example, $c=0, d=\sqrt{1-t}$ and let $a=b=\sqrt{t/2}.$ Then $\overline{a}b=t/2, |a|^2+|b|^2+|c|^2+|d|^2=1$
