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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. 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Dieses Zitat aus Hadwiger  1957] (S. 225) beschreibt recht gut die wesentlichen Elemente der Integralgeometrie: Es geht urn bewegte Figuren, also der Operation einer Gruppe unterworfene geometrische Objekte, und urn invariante Mittelwerte im Zusammenhang mit solchen bewegten Figuren. Integralgeometrie ist also ein Teilgebiet der Geometrie, das sich mit der Bestimmung und Anwendung von Mittelwerten geometrisch definierter Funk- tionen beziiglich invarianter Maf3e befaBt. Zu den Grundlagen der Integral- geometrie gehoren daher einerseits Teile der Theorie invarianter Maf3e auf topologischen Gruppen und homogenen Raumen, andererseits gewisse Ge- biete aus der Geometrie der Punktmengen, wie etwa der Polyeder, konvexen Mengen oder differenzierbaren Untermannigfaltigkeiten. 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