Gruber P. Convex and Discrete Geometry

VI Preface
theory of finite groups always was and still is of great importance. More recent is the
research on arrangements, matroids and the relations to graph theory.
The geometry of numbers, the elegant older sister of discrete geometry, was created
by Lagrange, Gauss, Korkin, Zolotarev, Fedorov, the leading figures Minkowski
and Voronoĭ, by Blichfeldt, by Delone, Ryshkov and the Russian school, by Siegel,
Hlawka, Schmidt, Davenport, Mahler, Rogers and others. Central problems of modern
research are the theory of positive definite quadratic forms, including reduction,
algorithmic questions and the lattice ball packing problem.
From tiny branches of geometry and number theory a hundred years ago, convexity,
discrete geometry and geometry of numbers developed into well-established
areas of mathematics. Now their doors are wide open to other parts of mathematics
and a number of applied fields. These include algebraic geometry, number theory,
in particular Diophantine approximation and algebraic number theory, theta series,
error correcting codes, groups, functional analysis, in particular the local theory of
normed spaces, the calculus of variations, eigenvalue theory in the context of partial
differential equations, further areas of analysis such as geometric measure theory,
potential theory, and also computational geometry, optimization and econometrics,
crystallography, tomography and mathematical physics.
We start with convexity in the context of real functions. Then convex bodies in
Euclidean space are investigated, making use of analytic tools and, in some cases, of
discrete and combinatorial ideas. Next, various aspects of convex polytopes are studied.
Finally, we consider geometry of numbers and discrete geometry, both from a
rather geometric point of view. For more detailed descriptions of the contents of this
book see the introductions of the individual chapters. Applications deal with measure
theory, the calculus of variations, complex function theory, potential theory, numerical
integration, Diophantine approximation, matrices, polynomials and systems of
polynomials, isoperimetric problems of mathematical physics, crystallography, data
transmission, optimization and other areas.
When writing the book, I became aware of the following phenomena in convex
and discrete geometry. (a) Convex functions and bodies which have certain properties,
have these properties often in a particularly strong form. (b) In complicated
situations, the average object has often almost extremal properties. (c) In simple
situations, extremal configurations are often regular or close to regular. The reader
will find numerous examples confirming these statements.
In general typical, rather than refined results and proofs are presented, even if
more sophisticated versions are available in the literature. For some results more
than one proof is given. This was done when each proof sheds different light on the
problem. Tools from other areas are used freely. The reader will note that the proofs
vary a lot. While in the geometry of numbers and in some more analytic branches of
convex geometry most proofs are crystal clear and complete, in other cases details
are left out in order to make the ideas of the proofs better visible. Sometimes we used
more intuitive arguments which, of course, can be made precise by inserting additional
detail or by referring to known results. The reader should keep in mind that all
this is typical of the various branches of convex and discrete geometry. Some proofs
are longer than in the original literature. While most results are proved, there are