Sails and Hilbert Bases

A Klein polyhedron is the convex hull of the nonzero integral points of a simplicial cone $C\subset\mathbb{R}^n$. There are relationships between these polyhedra and the Hilbert bases of monoids of integral points contained in a simplicial cone.
In the two-dimensional case, the set of integral points lying on the boundary of a Klein polyhedron contains a Hilbert base of the corresponding monoid. In general, this is not the case if the dimension is greater than or equal to three. However, in the three-dimensional case, we give a characterization of the polyhedra that still have this property. We give an example of such a sail and show that our criterion does not hold if the dimension is four.