Unifying framework for probabilistic and classical intersection theories

Develop a unifying framework that encompasses the relationship between the map from cycles to convex sets arising in probabilistic intersection theory and the map from subvarieties to classes of polytopes arising in toric intersection theory and the ring of conditions.

Background

The paper constructs a probabilistic intersection ring by associating submanifolds of compact Riemannian homogeneous spaces with Grassmann zonoids and then taking products of their classes. The authors observe that this construction is analogous to the polytope-algebra interpretation of toric intersection theory and to the ring of conditions of De Concini and Procesi.

A framework simultaneously encompassing these perspectives would clarify the common role of cycles, convex-geometric objects, and group actions in intersection theory. The paper explicitly states that no clear approach to establishing such a framework is currently evident, so the problem remains unresolved.