  In this paper, we study closed convex hulls of unitary orbits in various
C$^*$-algebras. For unital C$^*$-algebras with real rank zero and a faithful
tracial state determining equivalence of projections, a notion of majorization
describes the closed convex hulls of unitary orbits for self-adjoint operators.
Other notions of majorization are examined in these C$^*$-algebras. Combining
these ideas with the Dixmier property, we demonstrate unital, infinite
dimensional C$^*$-algebras of real rank zero and strict comparison of
projections with respect to a faithful tracial state must be simple and have a
unique tracial state. Also, closed convex hulls of unitary orbits of
self-adjoint operators are fully described in unital, simple, purely infinite
C$^*$-algebras.
