Convex polytopes are fundamental geometric objects
that have been investigated since antiquity. The beauty
of their theory is nowadays complemented by their
importance for many other mathematical subjects,
ranging from integration theory, algebraic topology and
algebraic geometry (toric varieties) to linear and
combinatorial optimization. In this chapter we try to
give a short introduction, provide a sketch of "what
polytopes look like" and "how they behave," with
many explicit examples, and briefly state some main
results (where further details are in the subsequent
chapters of this handbook). We concentrate on two main
topics: itemize} \item Combinatorial properties:
faces (vertices, edges, \ldots, facets) of polytopes
and their relations, with special treatments of the
classes of "lowdimensional polytopes" and "polytopes
with few vertices;" \item Geometric properties: volume
and surface area, mixed volumes and quermassintegrals,
including explicit formulas for the cases of the
regular simplices, cubes and crosspolytopes.
itemize} We refer to Grünbaum G} for a
comprehensive view of polytope theory, and to Ziegler
Z} and Schneider Schneider} for recent
treatments of the combinatorial resp. convex geometric
aspects of polytope theory.
