Minkowski additive operators under volume constraints

Abstract

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.

References in corpus (4)

• Crofton measures and Minkowski valuations
• Integral geometry of complex space forms
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• On the inverse Klain map