Gruber P. Convex and Discrete Geometry

244 Convex Polytopes
The material in this chapter is arranged as follows. After some preliminaries
and the introduction of the face lattice, combinatorial properties of convex polytopes
are considered, beginning with Euler’s polytope formula. In Sect. 14 we treat
the elementary volume as a valuation, and Hilbert’s third problem. Next, Cauchy’s
rigidity theorem for polytopal convex surfaces and rigidity of frameworks are discussed.
Then classical results of Minkowski, Alexandrov and Lindelöf are studied.
Lindelöf’s results deals with the isoperimetric problem for polytopes. Section 14
treats lattice polytopes, including results of Ehrhart, Reeve and Macdonald and the
Betke–Kneser valuation theorem. Applications of lattice polytopes deal with irreducibility
of polynomials and the Minding–Bernstein theorem on the number of
zeros of systems of polynomial equations. Finally we present an account of linear
optimization, including aspects of integer linear optimization.
For additional material the reader may wish to consult the books of Alexandrov
[16], Grünbaum [453], McMullen and Shephard [718], Brøndsted [171], Ewald
[315] and Ziegler [1045], the survey of Bayer and Lee [83] and other surveys in
the Handbooks of Convex Geometry [475] and Discrete and Computational Geometry
[476].
Regular polytopes and related topics will not be considered. For these we refer
to Coxeter [230, 232], Robertson [842], McMullen and Schulte [717] and Johnson
[551]. For McMullen’s algebra of polytopes, see [713].
14 Preliminaries and the Face Lattice
The simple concept of a convex polytope embodies a wealth of mathematical structure
and problems and, consequently, yields numerous results. The elementary theory
of convex polytopes deals with faces and normal cones, duality, in particular polarity,
separation and other simple notions. It was developed in the late eighteenth, the nineteenth
and the early twentieth century. Some of the results are difficult to attribute.
In part this is due to the large number of contributors.
In this section we first give basic definitions, and then show the equivalence of
the notions of V-polytopes and H-polytopes and, similarly, of V- and H-polyhedra.
We conclude with a short study of the face lattice of a convex polytope using polarity.
For more information, see the books cited earlier, to which we add Schneider
[907] and Schrijver [915].
14.1 Basic Concepts and Simple Properties of Convex Polytopes
In the following we introduce the notion of convex polytopes and describe two
alternative ways to specify convex polytopes: as convex hulls (V-polytopes) and as
intersections of halfspaces (H-polytopes). An example deals with a result of Gauss
on zeros of polynomials.