Symplectic Capacities of Domains Close to a Ball and Geodesics in the Space of Contact Forms

An old open question in symplectic geometry asks whether all normalised symplectic capacities coincide for convex domains in the standard symplectic vector space. I will show that this question has a positive answer for smooth convex domains which are C2

-close to a Euclidean ball. This is related to the question of existence of minimising geodesics in the space of contact forms on a closed contact manifold equipped with a Banach-Mazur-like metric. On the other hand, there are smooth domains which are arbitrary C1

-close to the ball for which the ball capacity is strictly smaller than the cylindrical capacity. This talk is based on recent joint work with Gabriele Benedetti and Oliver Edtmair.