Minkowski valuations under volume constraints

TL;DR

This paper characterizes Minkowski valuations with volume bounds, revealing only two operator types, and improves existing results on the difference body in convex geometry.

Contribution

It provides a complete description of volume-bounded Minkowski valuations, identifying two main operator families and enhancing known characterizations of the difference body.

Findings

Only two types of volume-bounded Minkowski valuations exist.

Identified operators are either cylindrical or 1-homogeneous.

Improved characterization results for the difference body.

Abstract

We provide a description of the space of continuous and translation invariant Minkowski valuations $\Phi:\mathcal{K}^n\to\mathcal{K}^n$ for which there is an upper and a lower bound for the volume of $\Phi(K)$ 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.