  We prove that Cuntz semigroups of C*-algebras satisfy Edwards' condition with
respect to every quasitrace. This condition is a key ingredient in the study of
the realization problem of functions on the cone of quasitraces as ranks of
positive elements. In the course of our investigation, we identify additional
structure of the Cuntz semigroup of an arbitrary C*-algebra and of the cone of
quasitraces.
