Generalised convexity with respect to families of affine maps

The standard convex closed hull of a subset of $\mathbb{R}^d$ 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. 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 above construction encompasses and generalises several known models in convex
stochastic geometry and allows us to gather them under a single umbrella. The talk is based on recent
works by Kalbuchko, Marynych, Temesvari, Thäle (2019), Marynych, Molchanov (2022) and Kabluchko,
Marynych, Molchanov (2023+).