  We investigate the closed convex hull of unitary orbits of selfadjoint
elements in arbitrary unital C*-algebras. Using a notion of majorization
against unbounded traces, a characterization of these closed convex hulls is
obtained. Furthermore, for C*-algebras satisfying Blackadar's strict comparison
of positive elements by traces or for collections of C*-algebras with a uniform
bound on their nuclear dimension, an upper bound for the number of unitary
conjugates in a convex combination required to approximate an element in the
closed convex hull within a given error is shown to exist. This property,
however, fails for certain "badly behaved" simple nuclear C*-algebras.
