Symplectic capacities are invariants that quantify the size of symplectic manifolds using themes from Hamiltonian dynamics and symplectic topology. While convexity is not preserved under symplectomorphisms, convex domains nevertheless exhibit notable behavior with respect to these capacities. Viterbo's volume-capacity conjecture (2000) suggests that, among convex domains of equal volume, the ball has maximal capacity. By capturing the interplay between convex and symplectic geometries, this simply formulated conjecture has become highly influential in the study of symplectic capacities, prompting extensive research. One result in this direction shows that smooth domains which are symplectic Zoll—a dynamical property—are local maximizers. In this talk, I will present a counterexample to Viterbo’s conjecture developed jointly with Yaron Ostrover and discuss follow-up questions. One implication is that a capacity maximizer cannot be smooth and strictly convex, raising the question of characterizing nonsmooth dynamical properties that detect local maximizers. I will propose a dynamical extension of the Zoll property to nonsmooth domains and discuss its equivalence with certain topological properties.
