Abstract: A conjecture of Claude Viterbo, from 1997, asserts that the symplectic capacity of a convex body should be bounded from above by its volume, when both are suitably normalized. This conjecture is of significant current interest, in part because its validity is now known to imply the Mahler conjecture in convex geometry. In this talk I will describe the proof of a weaker form of the inequality in which the role of the volume is played by a symplectic version of the mean-width. Time permitting, I will also describe some open problems suggested by this work.
