Open problem on Hampe’s intersection-ring polytope

Study the polytope obtained as the convex hull of all matroid classes in Hampe’s intersection ring, equivalently the convex hull of the Schubert-expansion points of loopless matroids with the corresponding loopless Schubert coordinates.

Background

Hampe’s intersection ring assigns classes only to loopless matroids. The paper explains that these classes expand into loopless Schubert matroid classes using the same coefficients as the Schubert indicator-function decomposition developed here. Consequently, Hampe’s open problem is identified with studying the convex hull of the corresponding points for loopless matroids, a polytope that the paper observes is a face of the labelled polytope.

References

In Section~6.2 of his paper introducing the intersection ring of matroids, Hampe left the following open problem: study the polytope that arises as the convex hull of all matroid classes in that ring.

— The polytope of all matroids  (2502.20157 - Ferroni et al., 27 Feb 2025) in Remark in Section 5, subsection “Simple and disconnected matroids”