  This paper is a study of the polyhedral geometry of Gelfand-Tsetlin patterns
arising in the representation theory $\mathfrak{gl}_n \C$ and algebraic
combinatorics. We present a combinatorial characterization of the vertices and
a method to calculate the dimension of the lowest-dimensional face containing a
given Gelfand-Tsetlin pattern.
  As an application, we disprove a conjecture of Berenstein and Kirillov about
the integrality of all vertices of the Gelfand-Tsetlin polytopes. We can
construct for each $n\geq5$ a counterexample, with arbitrarily increasing
denominators as $n$ grows, of a non-integral vertex. This is the first infinite
family of non-integral polyhedra for which the Ehrhart counting function is
still a polynomial. We also derive a bound on the denominators for the
non-integral vertices when $n$ is fixed.
