  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.
