Henk, Martin; Richter-Gebert, Jürgen; Ziegler, Günter M. Basic properties of convex polytopes. (English) Zbl 0911.52007 Goodman, Jacob E. (ed.) et al., Handbook of discrete and computational geometry. Boca Raton, FL: CRC Press. CRC Press Series on Discrete Mathematics and its Applications. 243-270 (1997). The authors try to give a short introduction to the theory of convex polytopes. They concentrate on the following two main topics: (i) combinatorial properties of faces, vertices, edges, …, facets of polytopes with special treatments of the classes of “low-dimensional polytopes” and “polytopes with few vertices”; (ii) geometric properties such as volume, surface area, and mixed volumes. The article is divided into following sections. 1. Combinatorial structure (\(d\)-polytope is introduced here as a convex hull of a finite set of points in \(\mathbb R^d\) and as a bounded solution set of a finite system of linear inequalities; the simplex, \(d\)-cube, and cross-polytope in \(\mathbb R^d\) are defined). 1.1. Faces (the theorem on face lattices of polytopes is given here). 1.2. Polarity . 1.3. Basis constructions. 1.4. More examples (zonotopes, cyclic polytopes, neighborly polytopes, and \((0,1)\)-polytopes are introduced here). 1.5. Three-dimensional polytopes and planar graphs (Steinitz’s theorem is formulated). 1.6. Four-dimensional polytopes and Schlegel diagrams (Richter-Gerbert’s universality theorem for 4-polytopes is formulated). 1.7. Polytopes with few vertices – Gale diagrams. 2. Metric properties. 2.1. Volume and surface area. 2.2. Mixed volumes (Schneider’s summation formula is given as well as McMullen’s formula for the volume of a special zonotope). 2.3. Quermassintegrals and intrinsic volumes. 3. Sources and related material (this section contains 35 references to related books and articles and gives a short description of some of them). Each section starts with a glossary containing all necessary definitions, presents (without proofs) basic theorems and formulas related to the topic under discussion, gives basic examples, and (sometimes) contains a list of open problems. The style of exposition is clear and elementary.For the entire collection see [Zbl 0890.52001]. Reviewer: V.Alexandrov (Novosibirsk) Cited in 32 Documents MSC: 52Bxx Polytopes and polyhedra Keywords:polytope; dimension; regular polytope; cube; hypercube; cross-polytope; support function; supporting hyperplane; outer normal vector; face; graded poset; atom; coatom; face lattice; combinatorial type; lattice; polarity; vertex figure; simple polytope PDFBibTeX XMLCite \textit{M. Henk} et al., in: Handbook of discrete and computational geometry. Boca Raton, FL: CRC Press. 243--270 (1997; Zbl 0911.52007) Online Encyclopedia of Integer Sequences: Triangle read by rows: T(n,k) is the number of Fibonacci binary words of length n and having k runs (0<=k<=n). A Fibonacci binary word is a binary word having no 00 subword. A run is a maximal sequence of consecutive identical letters.