  In convex geometry, the constructions that assign to a convex body its
difference body, projection body, or volume have the following properties: They
are (1) invariant under volume-preserving linear changes of coordinates; (2)
continuous; (3) finitely additive, and the resulting convex bodies are subsets
of an irreducible representation of the special linear group. In this paper we
explore the question whether there exist other constructions with these
properties. We discover a surprising dichotomy: There are no new examples if
one assumes translation invariance, but a plethora of examples without this
assumption.
