Examples related to Viterbo's conjectures

Viterbo conjectured that a normalized symplectic capacity, on convex domains of a given volume, is maximized for the ball. A stronger version of this conjecture asserts that all normalized symplectic capacities agree on convex domains. Since convexity is not symplectomorphism invariant, one can also ask to what extent these statements still hold for nonconvex domains. We survey some special cases and examples around these questions, including recent joint works with Julian Chaidez and Jean Gutt + Vinicius Ramos.