Alcoved polytopes are polytopes whose facets are normal to the $A_{n-1}$ root system. Polymatroids are polytopal analogues of matroids whose edges have specified directions. I will talk about joint work with Alex Postnikov on the class of polytopes, called polypositroids, that are simultaneously alcoved polytopes and polymatroids.

Our main motivation is that a matroid polytope is a polypositroid if and only if the matroid is a positroid, that is, it is the matroid of a point in the totally nonnegative Grassmannian. We will discuss: how to parametrize all polypositroids, how to take the polypositroid hull of a polymatroid, and the face structure of a polypositroid.