Diagonal Minkowski classes, zonoid equivalence, and stable laws

We consider the family of convex bodies obtained as transformations of a convex body $K$ by diagonal matrices, their Minkowski sums, and limits of them in the Hausdorff metric. We fix $K$ and consider these sums and their limits as 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 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.