A characterisation of zonoids

TL;DR

This paper characterizes zonoids by showing that a convex body is a zonoid if and only if all its associated spins are zonoids, providing a new geometric criterion.

Contribution

It introduces a novel characterization of zonoids based on the properties of their spins, linking local rotational invariance to global structure.

Findings

A convex body is a zonoid iff all its spins are zonoids.

Provides a new geometric criterion for identifying zonoids.

Connects local rotational invariance with the global zonoid property.

Abstract

Let $K$ be a unit ball of some norm in $R^n$. For an arbitrary direction $u\in R^n$, there is associated a unit-ball $K_u$, which is rotationally invariant with respect to rotations keeping $u$ fixed, called the $u$-spin of $K_u$. It is proved that $K$ is a zonoid if and only if all of its spins are zonoids.