On odd points and the volume of lattice polyhedra

Denote by _n³,n 2, the lattice consisting of all pointsx in ³ such thatnx belongs to the fundamental lattice ³ of points with integer coordinates. Letl_n be the subset of _n³ consisting of all points whose coordinates are odd multiples of 1/n. The purpose of this paper is to give several new Pick-type formulae for the volume of three-dimensional lattice polyhedra, that is, polyhedra with vertices in ³. Our formulae are in terms of numbers of only thel_n-points belonging to a lattice polyhedronP in contrast to already known formulae which employ numbers of all the _n³-points inP. On our way to establishing the formulae we show that the number of points froml_n belonging to a three-dimensional lattice polyhedronP has some polynomiality properties similar to those of the well-known Ehrhart polynomial expressing the number of points of _n³ inP. The paper contains also some comments on a problem of finding a volume formula which would employ only the setsl_n and which would be applicable to lattice polyhedra in arbitrary dimensions.Research partially supported by KBN Grant 2 P03A 008 10.