Minkowski Additive Operators Under Volume Constraints

We investigate Minkowski additive, continuous, and translation invariant operators \(\Phi :\mathcal {K}^n\rightarrow \mathcal {K}^n\) 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 \({{\mathrm{SO}}}(n)\)-equivariance, we obtain new characterization results for the difference body operator.