  We investigate Minkowski additive, continuous, and translation invariant
operators $\Phi:\mathcal{K}^n\to\mathcal{K}^n$ defined on the family of convex
bodies such that the volume of the image $\Phi(K)$ is bounded from above and
below by multiples of the volume of the convex body $K$, uniformly in $K$. We
obtain a representation result for an infinite subcone contained in the cone
formed by this type of operators. Under the additional assumption of
monotonicity or $SO(n)$-equivariance, we obtain new characterization results
for the difference body operator.
