The past 20 years have seen considerable progress and lively activity in various different areas of convex geometry. In order that this book still meet its intended purpose, it had to be updated and expanded. It remains the aim of the book to serve the newcomer to the field who wants an introduction from the very beginning, as well as the experienced reader who is either doing research in the field or is looking for some special result to be used elsewhere. In the introductory parts of the book, no greater changes have been necessary, but already here recent developments are reflected in a number of supplements. The main additions to the book are three new chapters, on valuations, on extensions and analogues of the Brunn–Minkowski theory, and on affine constructions and inequalities in the theory of convex bodies. The contents of Chapter 7 from the first edition are now found in Chapters 8 and 10 of the second edition, considerably extended. The structure of some other chapters has also been changed by, for example, dividing them into subsections, regrouping some material, or adding a new section. A few more technical proofs, which had been carried out in the first edition, have been replaced by hints to the original literature.

While the new topics added to the book all have their origins in the Brunn–Minkowski theory, their natural intrinsic developments may gradually have led them farther away. Proofs in this book are restricted to results which may have been basic for further developments, but are still close to the classical Brunn–Minkowski theory. In the remaining parts, we survey many recent results without giving proofs, but we always provide references to the sources where the proofs can be found. The section notes contain additional information.