Extreme points of graphing cycle-matroid base polytopes

Determine whether every extreme point of the basic minorizing-measure set associated with the cycle matroid of a graphing is exposed.

Background

The paper discusses a convex-analytic approach to measurable matroids based on minorizing charges and basic minorizing charges of a submodular rank function. In the finite case, extreme points of the corresponding polytopes are induced by independent sets and bases, but the structure of extreme points is substantially less understood in the measurable setting.

For the cycle matroid of a graphing, exposed points of the basic minorizing-measure set correspond to hyperfinite essential spanning forests. The unresolved issue is whether this characterization extends from exposed points to all extreme points, which would provide a more complete description of the convex structure underlying measurable graphing matroids.