  We prove that given a closed connected symplectic manifold equipped with a
Borel probability measure, an arbitrarily large portion of the measure can be
covered by a symplectically embedded polydisk, generalizing a result of
Schlenk. We apply this to constraints on symplectic quasi-states. Quasi-states
are a certain class of not necessarily linear functionals on the algebra of
continuous functions of a compact space. When the space is a symplectic
manifold, a more restrictive subclass of symplectic quasi-states was introduced
by Entov--Polterovich. We use our embedding result to prove that a certain
`soft' construction of quasi-states, which is due to Aarnes, cannot yield
nonlinear symplectic quasi-states in dimension at least four.
