The theory of mixed volumes provides a unified treatment of various important metric quantities in geometry, such as volume, surface area, and mean width. Apart from some historical roots in the works of Steiner [788] and Brunn [109], [110], its creation is due to Minkowski [624], [626]. The theory of area measures goes a step further, and can be regarded as a localization of the theory of mixed volumes. Area measures were introduced in the late 1930s, by Aleksandrov [2] and by Fenchel and Jessen [230], independently.

Until recently, there was no adequate introduction to these important topics in English, but fortunately, this situation has changed. The primary source of information is now Schneider's book [737], a superb sequel to Bonnesen and Fenchel's treatise [83]. (The latter is still well worth consulting, though it appeared too early for area measures to be included. It is regrettable that the books of Blaschke [71] and Hadwiger [370] have not yet been translated into English.) Apart from this, Webster's text [827, Chapter 6] provides an introduction to mixed volumes, and summaries of the theory are provided by Burago and Zalgaller [112, Chapter 4] and Sangwine-Yager [718].

Most of this appendix is also a summary, tailored to our particular requirements, of the theory of mixed volumes and area measures. The first section is designed to open the door to this enchanting but labyrinthine palace.