Gruber P. Convex and Discrete Geometry

310 Convex Polytopes
19 Lattice Polytopes
A convex lattice polytope in E d is the convex hull of a finite subset of the integer lattice
Z d . Equivalently, it is a convex polytope, all vertices of which are in Z d .LetP Z d
and P Z d p denote the spaces of all convex, resp. proper convex lattice polytopes in
E d . Lattice polytopes play a prominent role in convexity and several other branches
of mathematics, including the following:
Algebraic geometry (toric varieties, Newton polytopes)
Integer optimization
Tiling (Delone triangulations)
Crystallography
Combinatorial geometry (counting problems)
In this section we first present results of Ehrhart on lattice point enumerators and
study the relation of lattice point enumerators to the volume of lattice polytopes. In
particular, results of Pick, Reeve and Macdonald are proved. Next, we give a version
of Minkowski’s theorem on mixed volumes for integer linear combinations of
convex lattice polytopes due to McMullen and Bernstein. Finally, we present the theorem
of Betke–Kneser on valuations on the space of convex lattice polytopes which
is analogous to Hadwiger’s functional theorem on valuations on the space of convex
bodies. This result leads to short proofs of the lattice point enumeration theorems
of Ehrhart. For a different approach to enumeration problems for lattice polytopes
based on generating functions, we refer to Barvinok [80, 81]. Applications of lattice
polytopes deal with the irreducibility of polynomials in several variables and
the Minding–Kouchnirenko–Bernstein theorem on the number of zeros of a generic
system of polynomial equations.
Some of the proofs are rather complicated. The reader should not be misled by
a first look at the (much shorter) original proofs. In this section we use simple properties
of lattices and the Euler characteristic which are not specified in the pertinent
sections, but are easy to prove using the tools developed there. In addition, we use
simple material on Abelian groups. Tools on polynomials are proved as the proofs
are not easily available elsewhere. Let U denote the family of all integer unimodular
d × d matrices, i.e. d × d matrices with integer entries and determinant ±1.
For general information on lattice polytopes and lattice polyhedra, see the books
of Schrijver [915], Erdös, Gruber and Hammer [307], Gruber and Lekkerkerker
[447], Handelman [478], Barvinok [80] and Beck and Robins [86]. Schrijver, in
particular, studies convex lattice polyhedra in the context of integer optimization.
See also the surveys of McMullen and Schneider [716], Gritzmann and Wills [397],
Lagarias [625], Barvinok [81] and DeLoera [253] and the pertinent articles in the
collection on Integer Points in Polyhedra – Geometry, Number Theory, Algebra, Optimization
[536].
19.1 Ehrhart’s Results on Lattice Point Enumerators
The lattice point enumerators L, L o , L b : P Z d → Z are defined as follows, where #
denotes the counting function: