The aim of this book is to provide an introduction for students and
nonspecialists to a fascinating relation between combinatorial geometry
and algebraic geometry, as it has developed during the last two decades.
This relation is known as the theory of toric varieties or sometimes as
torus embeddings. Chapters I-IV provide a self-contained introduction to
the theory of convex poly- topes and polyhedral sets and can be used
independently of any applications to algebraic geometry. Chapter V forms
a link between the first and second part of the book. Though its
material belongs to combinatorial convexity, its definitions and
theorems are motivated by toric varieties. Often they simply translate
algebraic geometric facts into combinatorial language. Chapters VI-VIII
introduce toric va- rieties in an elementary way, but one which may not,
for specialists, be the most elegant. In considering toric varieties,
many of the general notions of algebraic geometry occur and they can be
dealt with in a concrete way. Therefore, Part 2 of the book may also
serve as an introduction to algebraic geometry and preparation for
farther reaching texts about this field. The prerequisites for both
parts of the book are standard facts in linear algebra (including some
facts on rings and fields) and calculus. Assuming those, all proofs in
Chapters I-VII are complete with one exception (IV, Theorem 5.1). In
Chapter VIII we use a few additional prerequisites with references from
appropriate texts.
