Convex Bodies: The Brunn–Minkowski Theory

A complete presentation of a central part of convex geometry, from basics for beginners, to the exposition of current research.

Now in its second edition, this classic text has been expanded to reflect significant developments in Brunn–Minkowski theory over the past two decades. It gives a complete presentation from basics to the exposition of current research, with full proofs, pointers to other fields and a fully updated reference list.At the heart of this monograph is the Brunn–Minkowski theory, which can be used to great effect in studying such ideas as volume and surface area and their generalizations. In particular, the notions of mixed volume and mixed area measure arise naturally and the fundamental inequalities that are satisfied by mixed volumes are considered here in detail. The author presents a comprehensive introduction to convex bodies, including full proofs for some deeper theorems. The book provides hints and pointers to connections with other fields and an exhaustive reference list. This second edition has been considerably expanded to reflect the rapid developments of the past two decades. It includes new chapters on valuations on convex bodies, on extensions like the Lp Brunn–Minkowski theory, and on affine constructions and inequalities. There are also many supplements and updates to the original chapters, and a substantial expansion of chapter notes and references.