Impressions of convexity - An illustration for commutator bounds

We determine the sharpest constant $C_{p,q,r}$ such that for all complex matrices $X$ and $Y$, and for Schatten $p$-, $q$- and $r$-norms the inequality $$\|XY-YX\|_p\leq C_{p,q,r}\|X\|_q\|Y\|_r$$ is valid. The main theoretical tool in our investigations is complex interpolation theory.