Symplectic Capacities of Domains Close to the Ball and Banach-Mazur Geodesics in the Space of Contact Forms

An old open question in symplectic topology is whether all normalized capacities coincide on convex bounded domains in the standard symplectic vector space. I will discuss this question for domains which are close to the Euclidean ball and its connection with the geometry of the space of contact forms with a Banach-Mazur pseudo-metric. This talk is based on a recent joint work with Gabriele Benedetti and Oliver Edtmair.