  The standard convex closed hull of a set is defined as the intersection of
all images, under the action of a group of rigid motions, of a half-space
containing the given set. In this paper we propose a generalisation of this
classical notion, that we call a $(K,\mathbb{H})$-hull, and which is obtained
from the above construction by replacing a half-space with some other convex
closed subset $K$ of the Euclidean space, and a group of rigid motions by a
subset $\mathbb{H}$ of the group of invertible affine transformations. The main
focus is put on the analysis of $(K,\mathbb{H})$-convex hulls of random samples
from $K$.
