Minkowski additive operators under volume constraints

TL;DR

This paper studies Minkowski additive operators on convex bodies that preserve volume constraints, providing new representations and characterizations, especially for the difference body operator under additional symmetry or monotonicity assumptions.

Contribution

It introduces a representation for a broad class of volume-constrained Minkowski additive operators and characterizes the difference body operator under symmetry or monotonicity.

Findings

Derived a representation for a subcone of volume-constrained Minkowski additive operators.

Provided new characterizations of the difference body operator with symmetry or monotonicity.

Established bounds relating the volume of the image to the original convex body.

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.