This chapter discusses the properties of mixed volumes emphasizing their geometric significance. A special class of mixed volumes is those involving an arbitrary convex body and a ball. The classical name for these is the quermass integrals, and they include volume, surface area, total mean curvature, and mean width. The study of mixed volumes has led immediately to the discovery of inequalities among them. The chapter describes mixed surface area measures and symmetrization. Mixed surface area measures that might be viewed as a local generalization of the mixed volumes, were introduced by Aleksandrov. It also presents an analysis of central unresolved problem in convexity and the general equality conditions for the Aleksandrov–Frenchel inequality. It discusses Bonnesen's inequality in the plane because it sharpens the Aleksandrov–Frenchel inequality or isoperimetric inequality.
