Bavard proved a duality theorem between commutator length and quasimorphisms.
Burago, Ivanov and Polterovich introduced the notion of a conjugation-invariant
norm which is a generalization of commutator length. Entov and Polterovich proved
Oh–Schwarz spectral invariants are subset-controlled quasimorphisms, which are
generalizations of quasimorphisms. We prove a Bavard-type duality theorem between
subset-controlled quasimorphisms on stable groups and conjugation-invariant
(pseudo)norms. We also pose a generalization of our main theorem and prove “stably
nondisplaceable subsets of symplectic manifolds are heavy” in a rough sense if that
generalization holds.
Keywords
Bavard's duality theorem, conjugation-invariant norm,
quasimorphism, heavy subset, stable nondisplaceability
