Let $X$ be a Banach space and $P$ be a projection in $B(X)$. Then $X$ can be renormed so that $P$ has norm $1$.
Can the same be done for a family of projections? That is, given finitely many projections $P_1$,...,$P_n$ in $B(X)$ is there an equivalent norm under which all projections become contractive?
Are there any assumptions other than $P_i$ commuting that would ensure this?