Impressions of convexity: An illustration for commutator bounds

We determine the sharpest constant Cp,q,rCp,q,r such that for all complex matrices XX and YY, and for Schatten pp-, qq- and rr-norms the inequality‖XY-YX‖p⩽Cp,q,r‖X‖q‖Y‖r‖XY-YX‖p⩽Cp,q,r‖X‖q‖Y‖ris valid. The main theoretical tool in our investigations is complex interpolation theory.