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].

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)

##### MSC:

52Bxx | Polytopes and polyhedra |