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## Über dieses Buch

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.

## Inhaltsverzeichnis

### I. Convex Bodies

Abstract
Most of the sets considered in the first part of the book are subsets of Euclidean n-space. Many definitions and theorems could be stated in an affinely invariant manner. We do not, however, stress this point. If we use the symbol ℝ n , it should be clear from the context whether we mean real vector space, real affine space, or Euclidean space. In the latter case, we assume the ordinary scalar product
$$\left\langle {x,y} \right\rangle = \xi _1 \eta _1 + ... + \xi _n \eta _n$$
so that the square of Euclidean distance between points x and y equals
$$||x - y||^2 = \left\langle {x - y,x - y} \right\rangle .$$
Günter Ewald

### II. Combinatorial theory of polytopes and polyhedral sets

Abstract
We will turn now to the specific properties of convex polytopes or, briefly, poly-topes. They have been introduced in I.1 as convex hulls of finite point sets in ℝ n . Our first aim is to show that, equivalently, convex polytopes can be defined as bounded intersections of finitely many half-spaces. (This fact is of particular relevance in linear optimization).
Günter Ewald

### III. Polyhedral spheres

Abstract
In the preceding chapter, we dealt with boundary complexes of convex polytopes. They consist of cell decompositions of topological spheres, the cells again being convex polytopes. If, however, any cell decomposition of a topological sphere is given, there need not exist a convex polytope with isomorphic (in the sense of inclusion of cells) boundary complex. We shall present counter-examples in section 4 below. In fact, one of the major unsolved problems in convex polytope theory is to find necessary and sufficient conditions for a cell-composed sphere to be isomorphic to the boundary complex of a polytope (Steinitz problem).
Günter Ewald

### IV. Minkowski sum and mixed volume

Abstract
A fundamental operation for convex sets is the following (which can be defined for arbitrary sets in ℝ n ).
Günter Ewald

### V. Lattice polytopes and fans

Abstract
In I,1, we introduced the (polyhedral) cone σ as the positive hull of finitely many vectors x1,..., x k .
Günter Ewald

### VI. Toric Varieties

Abstract
From its beginning, algebraic geometry is concerned with sets of zeros of finitely many polynomials. These affine algebraic sets form a basic part of the theory, usually as “charts” of which more general varieties are built up (by “gluing together”). The underlying field of coefficients may be general or restricted to one of the fields ℚ, ℝ, ℂ of rational, real, or complex numbers, depending on the topic discussed and the methods used.
Günter Ewald

### VII. Sheaves and projective toric varieties

Abstract
In VI, Lemma 1.27, we introduced rational functions as functions whose restriction on an appropriate Zariski open set U0 is regular, that is, represented by a quotient f = g/h of polynomials g,h with h nowhere 0 on U0. Even more concretely, we may choose U0 to be a Zariski open subset of the torus T so that the rational functions on XΣ are all given by rational functions on T.
Günter Ewald

### VIII. Cohomology of toric varieties

Abstract
In this last chapter, we wish to study topological properties of toric varieties which are hidden in the structure of the Chow ring.
Günter Ewald

### Backmatter

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