## Daniel Dörfler, Andreas Löhne, Convex sets approximable as the sum of a compact set and a cone

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DOI: 10.23952/jnva.8.2024.4.12

Volume 8, Issue 4, 1 August 2024, Pages 681-689

**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 hyperbolic sets, i.e., convex sets contained in the sum of their recession cone and a compact convex set 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.

**How to Cite this Article**:

D. Dörfler, A. Löhne, Convex sets approximable as the sum of a compact set and a cone, J. Nonlinear Var. Anal. 8 (2024), 681-689.