  This paper establishes an aspect of Bohr's correspondence principle, i.e.
that quantum mechanics converges in the high frequency limit to classical
mechanics, for commuting semiclassical unitary operators. We prove, under
minimal assumptions, that the semiclassical limit of the convex hulls of the
quantum spectrum of a collection of commuting semiclassical unitary operators
converges to the convex hull of the classical spectrum of the principal symbols
of the operators.
