In this talk, we will study the semialgebraic set of linear subspaces of fixed dimension that intersect a given polytope. This set can be described as a union of cells in the complement of a Schubert arrangement associated with the polytope, within the Grassmannian. In particular, we will focus on subspaces that intersect a simplicial polytope and explore its connections to other well-studied semialgebraic subsets of the Grassmannian: the totally non-negative Grassmannian and the (loop) amplituhedron.
Based on joint work with Sebastian Seemann.