Generalised convexity with respect to families of affine maps

TL;DR

This paper introduces a generalized convex hull concept called $(K,\mathbb{H})$-hull, extending classical convexity by replacing half-spaces and rigid motions with arbitrary convex sets and affine transformations, and analyzes their properties for random samples.

Contribution

It proposes a new generalized convexity framework called $(K,\mathbb{H})$-hull, broadening classical convex hulls with affine transformation groups and convex sets, and studies their properties for random samples.

Findings

Introduced the $(K,\mathbb{H})$-hull as a generalization of classical convex hulls.

Analyzed the properties of $(K,\mathbb{H})$-convex hulls for random samples.

Provided insights into the structure and behavior of these generalized convex hulls.

Abstract

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 $K$.