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.