Limit Theorems for Herer Expectations of Random Sets in Banach Spaces

Abstract A law of large numbers and a central limit theorem are proved for the Herer expectation of integrably bounded random closed sets in a separable Banach space. These results are obtained in an appropriate metric betweeen sets which is constructed from the norm. Although, in a given space, it is generally weaker than the Hausdorff metric, these results include in many cases (via suitable renormings or embeddings) the corresponding limit theorems for the Aumann expectation in the Hausdorff metric. The results rely on new representation results for the Herer expectation as an intersection of halfspaces, which strongly depend on the geometry of the dual unit ball.