On symplectic capacities and their blind spots

In this paper, we settle three basic questions concerning the Gutt–Hutchings capacities from [11] which are conjecturally equal to Ekeland–Hofer [7, 8]. Our primary result settles a version of recognition question [4], in negative. We prove that together with volume do not constitute complete set symplectic invariants for star-shaped domains smooth boundary. particular, construct family such domains, all have same and volume, but no two symplectomorphic. also establish independence properties capacities. that, even boundaries, these independent by constructing whose agree volumes differ. mutually constructing, any [Formula: see text], boundary text]th. The constructions underlying results exotic. They convex concave toric as defined [11], where authors beautiful formulae their A key progress made here is significant simplification under an additional symmetry assumption. This allows us identify new blind spots used desired examples. yields explicit computations many