P. Terán (2010). Journal of Convex Analysis 17, 277-292.
Although there is no probability in it, this paper is in fact part of a strand of research in random set theory. The driving question is as follows: what is the relationship between the Aumann and the Herer expectation of a random set? Since Aumann's definition relies on integration in Banach spaces whereas Herer's is geometric, the mere fact that definite relationships exist is interesting.
I proved that both expectations are identical if and only if the space has the Mazur Intersection Property that every closed bounded convex set can be written as the intersection of a family of balls (this happened in a paper in Test which, I realize now, I haven't uploaded here yet).
However, to reply the very next question (`But are they related at all when they are not identical?') I needed to study some properties of those sets which can be written as intersections of balls, and of the `ball hull', the analog of the convex hull in this context. Which is what this paper is about.
All the reasoning in the paper is elementary so it makes a nice starting point for further research.
Up the line:
·On the equivalence of Aumann and Herer expectations of random sets (2008).
Down the line:
·Expectations of random sets in Banach spaces (accepted 2013).
To download, click on the title or here.