Let denote an integral convex Polytope of Dimension in a lattice , and let
denote the number of Lattice Points in dilated by a factor of the integer ,

(1) |

(2) |

- 1. is the Content of .
- 2. is half the sum of the Contents of the -D faces of .
- 3. .

(3) |

(4) |

(5) |

**References**

Ehrhart, E. ``Sur une problème de géométrie diophantine linéaire.'' *J. Reine angew. Math.*
**227**, 1-29, 1967.

MacDonald, I. G. ``The Volume of a Lattice Polyhedron.'' *Proc. Camb. Phil. Soc.* **59**, 719-726, 1963.

McMullen, P. ``Valuations and Euler-Type Relations on Certain Classes of Convex Polytopes.'' *Proc. London
Math. Soc.* **35**, 113-135, 1977.

Pommersheim, J. ``Toric Varieties, Lattices Points, and Dedekind Sums.'' *Math. Ann.* **295**, 1-24, 1993.

Reeve, J. E. ``On the Volume of Lattice Polyhedra.'' *Proc. London Math. Soc.* **7**, 378-395, 1957.

Reeve, J. E. ``A Further Note on the Volume of Lattice Polyhedra.'' *Proc. London Math. Soc.* **34**,
57-62, 1959.

© 1996-9

1999-05-25