Convex sets approximable as the sum of a compact set and a cone
Abstract
The class of convex sets that admit approximations as Minkowski sum of a compact convex set and a closed convex cone in the Hausdorff distance is introduced. These sets are called approximately Motzkin-decomposable and generalize the notion of Motzkin-decomposability, i.e. the representation of a set as the sum of a compact convex set and a closed convex cone. We characterize these sets in terms of their support functions and show that they coincide with self-bounded sets, i.e. sets contained in the sum of a compact convex set and a closed convex cone, if their recession cones are polyhedral but are more restrictive in general. In particular we prove that a set is approximately Motzkin-decomposable if and only if its support function has a closed domain relative to which it is continuous.
Keywords
Cite
@article{arxiv.2401.09026,
title = {Convex sets approximable as the sum of a compact set and a cone},
author = {Daniel Dörfler and Andreas Löhne},
journal= {arXiv preprint arXiv:2401.09026},
year = {2024}
}
Comments
12 pages, 1 figure; add Remark 3.12, acknowledgements and 4 references, replace the term self-bounded with hyperbolic