Suppose $A$ and $B$ have the same spectral radius $\rho$. We can find a norm $\| \cdot \|_A $ s.t. $\|A\|_A - \epsilon < \rho$. We can likewise find a another norm s.t. $\|B\|_B - \epsilon < \rho$. Under what conditions can we find a norm $\|\cdot\|_*$ such that both
$$\|A\|_* - \epsilon < \rho $$
and
$$\|B\|_* - \epsilon < \rho$$
Having done some digging it would seem one sufficient condition would be for $A$ and $B$ to be normal and another for the matrices to be simultaneously diagonalizable. Is there a necessary and sufficient condition?