Diagonal Minkowski classes, zonoid equivalence, and stable laws

TL;DR

This paper explores a family of convex bodies generated by diagonal transformations and Minkowski sums, establishing conditions for the injectivity of an integral transform called the K-transform, with applications to stable laws and geometric analysis.

Contribution

It generalizes known properties of zonoids and cosine transform to a broader class of convex bodies via the K-transform, linking geometric and probabilistic concepts.

Findings

The K-transform's injectivity is characterized for generalised zonoids.

Equivalence of density of support functions and injectivity of the K-transform is established.

Connections between convex geometry and stable laws are elucidated.

Abstract

We consider the family of convex bodies obtained from an origin symmetric convex body $K$ by multiplication with diagonal matrices, by forming Minkowski sums of the transformed sets, and by taking limits in the Hausdorff metric. Support functions of these convex bodies arise by an integral transform of measures on the family of diagonal matrices, equivalently, on Euclidean space, which we call $K$-transform. In the special case, if $K$ is a segment not lying on any coordinate hyperplane, one obtains the family of zonoids and the cosine transform. In this case two facts are known: the vector space generated by support functions of zonoids is dense in the family of support functions of origin symmetric convex bodies; and the cosine transform is injective. We show that these two properties are equivalent for general $K$. For $K$ being a generalised zonoid, we determine conditions that ensure the injectivity of the $K$-transform. Relations to mixed volumes and to a geometric description of one-sided stable laws are discussed. The later probabilistic application gives rise to a family of convex bodies obtained as limits of sums of diagonally scaled $\ell_p$-balls.