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We prove that all normalized symplectic capacities coincide on smooth domainsin $\mathbb C^n$ which are $C^2$-close to the Euclidean ball, whereas thisfails for some smooth domains which are just $C^1$-close to the ball. We alsoprove that all symplectic capacities whose value on ellipsoids agrees with thatof the $n$-th Ekeland-Hofer capacity coincide in a $C^2$-neighborhood of theEuclidean ball of $\mathbb C^n$. These results are deduced from a generaltheorem about contact forms which are $C^2$-close to Zoll ones, saying thatthese contact forms can be pulled back to suitable "quasi-invariant" contactforms. We relate all this to the question of the existence of minimizinggeodesics in the space of contact forms equipped with a Banach-Mazurpseudo-metric. Using some new spectral invariants for contact forms, we provethe existence of minimizing geodesics from a Zoll contact form to any contactform which is $C^2$-close to it. This paper also contains an appendix in whichwe review the construction of exotic ellipsoids by the Anosov-Katok conjugationmethod, as these are related to the above mentioned pseudo-metric.

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