  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 that Oh-Schwarz spectral invariants are subset-controlled quasimorphisms
which are geralizations of quasimorphisms. In the present paper, we prove a
Bavard-type duality theorem between conjugation-invariant (pseudo-)norms and
subset-controlled quasimorphisms on stable groups. %We also give an application
of its generalization to symplectic geometry related to Entov-Polterovich's
theory on heavy subsets. We also pose a generalization of our main theorem and
prove that "stably non-displaceable subsets of symplectic manifolds are heavy"
in a rough sense if that generalization holds.
