Convex Polytopes The early history of convex polytopes is lost. About 2000 BC convex polytopes appeared in a mathematical context in the Sumerian civilization, in Babylonia and in Egypt. Sources are the Moscow papyrus and the Rhind papyrus. Some of the regular polytopes were already known by then. A basic problem was to calculate the vol- umes of truncated pyramids. This was needed to determine the number of bricks for fortifications and buildings. Babylonians sometimes did the calculations correctly, sometimes not, while Egyptians used the right formula. For pertinent information the author is obliged to the assyriologist Hermann Hunger [531]. In the fifth century BC Democritos also discovered this formula and Eudoxos proved it, using the method of exhaustion. Theaitetos developed a theory of regular polytopes, later treated by Plato in the dialogue Timaios. Euclid, around 300 BC, considered metric proper- ties of polytopes, the volume problem, including the exhaustion method, and the five regular polytopes, the Platonic solids. Zenodoros, who lived sometime between 200 BC and 90 AD, studied the isoperimetric problem for polygons and polytopes and Pappos, about 300 AD, dealt with the semi-regular polytopes of Archimedes. In the renaissance the study of convex polytopes was in the hands of artists such as Uccello, Pacioli, da Vinci, D¨urer, and Jamnitzer. Then it went back to mathe- matics. Kepler investigated the regular and the semi-regular polytopes and planar tilings. Descartes considered convex polytopes from a metric point of view, almost arriving at Euler’s polytope formula, discovered by Euler only hundred years later. Contributions to polytope theory in the late eighteenth and the nineteenth century are due to Legendre, Cauchy, Steiner, Schlafli¨ and others. At the turn of the nine- teenth and in the twentieth century important results were given by Minkowski, Dehn, Sommerville, Steinitz, Coxeter and numerous contemporaries. At present, emphasis is on the combinatorial, algorithmic, and algebraic aspects. Modern rela- tions to other areas date back to Newton (polynomials), Fourier (linear optimization), Dirichlet, Minkowski and Vorono˘ı(quadraticforms)andFedorov(crystallography). In recent decades polytope theory was strongly stimulated and, in part, re-oriented by linear optimization, computer science and algebraic geometry. Polytope theory, in turn, had a certain impact on these areas. For the history, see Federico [318] and Malkevitch [681]. 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 poly- topes 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 dis- cussed. Then classical results of Minkowski, Alexandrov and Lindelof¨ are studied. Lindelof’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 irre- ducibility 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 Alexan- drov [16], Gr¨unbaum [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 Geome- try [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 struc- ture 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 nine- teenth 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-andH-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. 14 Preliminaries and the Face Lattice 245 Convex Polytopes and Faces A convex polytope P in Ed is the convex hull of a finite, possibly empty, set in d E .IfP conv x1,...,xn ,thentheextremepointsofP are among the points = { } x1,...,xn by Minkowski’s theorem 5.5 on extreme points. Thus, a convex polytope has only finitely many extreme points. If, conversely, a convex body has only finitely many extreme points, then it is the convex hull of these, again by Minkowski’s theo- rem. Hence, the convex polytopes are precisely the convex bodies with finitely many extreme points. The intersection of P with a support hyperplane H is a face of P.It is not difficult to prove that (1) H P conv(H x1,...,xn ). ∩ = ∩{ } This shows that a face of P is again a convex polytope. The faces of dimension 0 are the vertices of P.Thesearepreciselytheextreme,actuallytheexposedpointsofP. The faces of dimension 1 are the edges of P and the faces of dimension dim P 1the facets.Theemptyset and P itself are called the improper faces,allotherfacesare− ∅ proper.SinceafaceofP is the convex hull of those points among x1,...,xn which are contained in it, P has only finitely many faces. Since each boundary point of P is contained in a support hyperplane by Theorem 4.1, bd P is the union of all proper faces of P.ByF(P) we mean the family of all faces of P,including and P.The d d ∅ d space of all convex polytopes in E is denoted by (E ) and p p(E ) is its sub-space consisting of all proper convex polytopes,P = P that is thoseP with= non-emptyP interior. Gauss’s Theorem on the Zeros of the Derivative of a Polynomial In an appendix to his third proof of the fundamental theorem of algebra, Gauss [363] proved the following result. Theorem 14.1. Let p be a polynomial in one complex variable. Then the zeros of its derivative p$ are contained in the convex polygon determined by the zeros of p. Proof. Let z1,...,zn C be the zeros of p,eachwrittenaccordingtoitsmultiplic- ity. Then ∈ p(z) a(z z1) (z zn) for z C = − ··· − ∈ with suitable a C.Letz z1,...,zn.Dividingthederivativep$(z) by p(z) implies that ∈ &= p$(z) 1 1 z z1 z zn (2) ¯ −¯ 2 ¯ −¯ 2 . p(z) = z z1 +···+ z zn = z z1 +···+ z zn − − | − | | − | Assume now that z is a zero of p .Wehavetoshowthatz conv z1,...,zn .Ifz $ ∈ { } is equal to one of z1,...,zn,thisholdstrivially.Otherwise(2)showsthat 1 1 2 z1 2 zn z z1 z zn | − | +···+ | − | z 1 1 . = 2 2 z z1 z zn | − | +···+ | − | Thus z is a convex combination of z1,...,zn. '( 246 Convex Polytopes V-Polytopes and H-Polytopes Convex polytopes as defined earlier are also called convex V-polytopes.HereV stands for vertices. Dually, a convex H-polyhedron is the intersection of finitely many closed halfspaces. A bounded convex H-polyhedron is called a convex H-polytope. AformalproofofthefollowingfolktheoremisduetoWeyl[1021],seealso Minkowski [744], Sect. 4. Theorem 14.2. Let P Ed .Thenthefollowingstatementsareequivalent: ⊆ (i) PisaconvexV-polytope. (ii) PisaconvexH-polytope. Proof. (i) (ii) We may suppose that dim P d. P has only finitely many faces. Since each⇒ boundary point of P is contained in= a support hyperplane of P by Theo- rem 4.1, bd P is the union of its proper faces. Connecting an interior point of P with alinesegmentwhichmissesallfacesofdimensionatmostd 2withanexterior point, each point where it intersects the boundary of P must be− contained in a face of dimension d 1, i.e. in a facet. Thus P must have facets. Let Hi , i 1,...,m, − = be the hyperplanes containing the facets of P and Hi− the corresponding support halfspaces. We claim that (3) P H − H −. = 1 ∩···∩ m The inclusion P H − H is trivial. To show the reverse inclusion, let ⊆ 1 ∩···∩ m− x Ed P.ForeachofthefinitelymanyfacesofP of dimension at most d 2, consider∈ \ the affine hull of the face and x.Chooseapointy int P which is contained− in none of these affine hulls. The intersection of the line∈ segment x, y with bd P then is a point z bd P which is contained in none of these affine hulls[ ] and thus in none of the faces∈ of P of dimension at most d 2. Since bd P is the union of all − faces, z is contained in a suitable facet and thus in one of the hyperplanes, say Hi . Then x Hi− and therefore x H1− Hm−.HenceP H1− Hm−, concluding&∈ the proof of (3). &∈ ∩···∩ ⊇ ∩···∩ (ii) (i) Let P H − H be bounded, where each H −, i 1, , m, ⇒ = 1 ∩···∩ m− i = ··· is a halfspace with boundary hyperplane Hi . Clearly, P is a convex body. By Minkowski’s theorem 5.5, P is the convex hull of its extreme points.