We choose $I = [-1,1]$ and write $C_{n,m}$ instead of $C_{n,m}^I$. We fix an extremal pair $(Q_0, R_0)$, i.e., $Q_0$ and $R_0$ are monic and $\frac{\|Q_0\|_I \|R_0\|_I}{\|Q_0 R_0\|_I} = C_{n,m}$.
Let $J$ be a segment contained in $I$ such that $\left\|Q_0\right\|_J = \left\|Q_0\right\|_I$ and $\left\|R_0\right\|_J = \left\|R_0\right\|_I$. Show that:
$$\left\|Q_0 R_0\right\|_J = \left\|Q_0 R_0\right\|_I$$