Let $T_1,\dots,T_n$ be bounded linear operators on a complex Hilbert space $H$. We study the question whether it is possible to find a unit vector $x\in H$ such that $|\langle T_jx, x\rangle|$ is large for all $j$. Thus we are looking for a generalization of the well-known fact for $n = 1$ that the numerical radius $w(T)$ of a single operator T satisfies $w(T)\ge \|T\|/2$.