The upper bound is \(c_n(I) \le \| I\| \le 1\). For the lower bound let \(N \subseteq \ell _1\) be closed of codimension \(\le n\). Then \(\ell _1/N\) is finite dimensional, so among the classes \([e_i]\) of the unit vectors two are arbitrarily close (Bolzano–Weierstrass). Lifting the difference gives \(x \in N\) close to \(e_i - e_{i'}\), with \(\| x\| _1 \approx 2\) but \(\| I x\| _\infty \approx 1\); hence \(\| I|_N\| \ge \tfrac 12 - \varepsilon \) for every \(\varepsilon {\gt} 0\).