Polynomials and spatial Pick-type theorems

Pick's theorem about the area of a simple lattice planar polygon has many extensions and generalizations even in the planar case. The theorem has also higher-dimensional generalizations, which are not as commonly known as the 2-dimensional case. The aim of the paper is, on one hand, to give a few new higher-dimensional generalizations of Pick's theorem and, on the other hand, collect known ones. We also study some relationships between lattice points in a lattice polyhedron which lead to some new Pick-type formulae. Another purpose of this paper is to pose several problems related to the subject of higher-dimensional Pick-type theorems. We hope that the paper may popularize the idea of determining the volume of a lattice polyhedron P by reading information contained in a lattice and the tiling of the space generated by the lattice.