In the 1960s, Henry Crapo defined the beta invariant, $\beta$, for a matroid $M$. This nonnegative integer is ubiquitous in the study of matroids. It is related to the chromatic polynomial of $M$, the Tutte polynomial of $M$, the connectedness of $M$, and more. The first concrete enumerative interpretation of $\beta$ was in computing the number of relatively bounded regions of a hyperplane arrangement, attributed to Thomas Zaslavsky.
While $\beta$ itself is well-studied, one might ask if it has a promising analogy in the realm of real-valued polymatroids. In this talk, I will present such a candidate, which I hope to convince you is a natural extension (although not quite a generalization) of $\beta$ into the polymatroid world, and I will show you one of its geometric applications.