Affine Minkowski valuations

In geometric valuation theory, well-known examples of Minkowski valuations intertwining the special linear group are defined by the difference operator, projection operator and the moment operator. While the difference and projection operator are translation invariant the moment operator is not. The former examples can be seen as maps with values in the set of convex bodies in the first and the (n-1)-th exterior power of R^n respectively. By a result of Monika Ludwig the difference operator and the projection operator are the only continuous, translation invariant Minkowski valuations with their corresponding codomain that commute with the special linear group. We ask whether there is also a Minkowski valuation satisfying all of these properties but whose codomain is the set of convex bodies in the k-th exterior power of R^n for 1 < k < n. We give an answer to a more general question. We prove that for any finite-dimensional irreducible SL(n)-representation W such a Minkowski valuation with values in the set of convex bodies in W exists if and only if W equals the set of real numbers, the first or the (n-1)-th exterior power of R^n. Finally, we give some new examples satisfying all properties mentioned above but translation invariance. If n < 4 we show that there is a continuous and SL(n) equivariant Minkowski valuation defined on the set of convex bodies containing the origin in its interior with values in the set of convex bodies in W, for any finite-dimensional SL(n)-representation W. One of these examples is a generalization of the moment operator. The existence of a Busemann-Petty type inequality for the generalized moment operator is discussed