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.

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