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In convex geometry, the constructions that assign to a convex body itsdifference body, projection body, or volume have the following properties: Theyare (1) invariant under volume-preserving linear changes of coordinates; (2)continuous; and (3) finitely additive. In this paper we explore the questionwhether there exist other constructions with these properties. We discover asurprising dichotomy: There are no new examples if one assumes translationinvariance, but a plethora of examples without this assumption.

Valuations in affine convex geometry