Gruber P. Convex and Discrete Geometry

Preface
In this book we give an overview of major results, methods and ideas of convex and
discrete geometry and their applications. Besides being a graduate-level introduction
to the field, the book is a practical source of information and orientation for convex
geometers. It should also be of use to people working in other areas of mathematics
and in the applied fields.
We hope to convince the reader that convexity is one of those happy notions of
mathematics which, like group or measure, satisfy a genuine demand, are sufficiently
general to apply to numerous situations and, at the same time, sufficiently special to
admit interesting, non-trivial results. It is our aim to present convexity as a branch of
mathematics with a multitude of relations to other areas.
Convex geometry dates back to antiquity. Results and hints to problems which are
of interest even today can already be found in the works of Archimedes, Euclid and
Zenodorus. We mention the Platonic solids, the isoperimetric problem, rigidity of
polytopal convex surfaces and the problem of the volume of pyramids as examples.
Contributions to convexity in modern times started with the geometric and analytic
work of Galileo, the Bernoullis, Cauchy and Steiner on the problems from antiquity.
These problems were solved only in the nineteenth and early twentieth century by
Cauchy, Schwarz and Dehn. Results without antecedents in antiquity include Euler’s
polytope formula and Brunn’s inequality. Much of modern convexity came into being
with Minkowski. Important later contributors are Blaschke, Hadwiger, Alexandrov,
Pogorelov, and Klee, Groemer, Schneider, McMullen together with many further
living mathematicians. Modern aspects of the subject include surface and curvature
measures, the local theory of normed spaces, best and random approximation,
affine-geometric features, valuations, combinatorial and algebraic polytope theory,
algorithmic and complexity problems.
Kepler was the first to consider problems of discrete geometry, in particular packing
of balls and tiling. His work was continued by Thue, but the systematic research
began with Fejes Tóth in the late 1940s. The Hungarian school deals mainly with
packing and covering problems. Amongst numerous other contributors we mention
Rogers, Penrose and Sloane. Tiling problems are a classical and also a modern topic.
The ball packing problem with its connections to number theory, coding and the