  The Groebner fan of an ideal $I\subset k[x_1,...,x_n]$, defined by Mora and
Robbiano, is a complex of polyhedral cones in $R^n$. The maximal cones of the
fan are in bijection with the distinct monomial initial ideals of $I$ as the
term order varies. If $I$ is homogeneous the Groebner fan is complete and is
the normal fan of the state polytope of $I$. In general the Groebner fan is not
complete and therefore not the normal fan of a polytope. We may ask if the
restricted Groebner fan, a subdivision of $R_{>=0}^n$, is regular i.e. the
normal fan of a polyhedron. The main result of this paper is an example of an
ideal in $Q[x_1,...,x_4]$ whose restricted Groebner fan is not regular.
