Generalised convexity with respect to families of affine maps

Source abstract

Abstract The standard closed convex 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 , ℍ)-hull, and which is obtained from the above construction by replacing a half-space with some other closed convex subset K of the Euclidean space, and a group of rigid motions by a subset ℍ of the group of invertible affine transformations. The main focus is on the analysis of ( K , ℍ)-convex hulls of random samples from K .