Symplectic capacities of domains close to the ball and Banach–Mazur geodesics in the space of contact forms

We prove that all normalized symplectic capacities coincide on smooth domains in Cⁿ which are C ²-close to the Euclidean ball, whereas this fails for some smooth domains which are just C ¹-close to the ball. We also prove that all symplectic capacities whose value on ellipsoids agrees with that of the nth Ekeland–Hofer capacity coincide in a C ²-neighborhood of the Euclidean ball of Cⁿ. These results are deduced from a general theorem about contact forms which are C ²-close to Zoll ones, saying that these contact forms can be pulled back to suitable quasi-invariant contact forms. We relate all this to the question of the existence of minimizing geodesics in the space of contact forms equipped with a Banach–Mazur pseudometric. Using some new spectral invariants for contact forms, we prove the existence of minimizing geodesics from a Zoll contact form to any contact form which is C ²-close to it. This paper also contains an appendix in which we review the construction of exotic ellipsoids by the Anosov–Katok conjugation method, as these are related to the above-mentioned pseudometric.