Minkowski valuations under volume constraints

We provide a description of the space of continuous and translation invariant Minkowski valuations Φ:Kn→Kn for which there is an upper and a lower bound for the volume of Φ(K) in terms of the volume of the convex body K itself. Although no invariance with respect to a group acting on the space of convex bodies is imposed, we prove that only two types of operators appear: a family of operators having only cylinders over (n−1)-dimensional convex bodies as images, and a second family consisting essentially of 1-homogeneous operators. Using this description, we give improvements of some known characterization results for the difference body.