Convex Geometry of Subword Complexes

Steinitz's problem asks whether a triangulated sphere is realizable geometrically as the boundary of a convex polytope. The determination of the polytopality of subword complexes is a resisting instance of Steinitz's problem. Indeed, since their creation more than 15 years ago, subword complexes built up a wide portfolio of relations and applications to many other areas of research (Schubert varieties, cluster algebras, associahedra, tropical Grassmannians, to name a few) and a lot of efforts has been put into realizing them as polytopes, with little success. In this talk, I will present some reasons why this problem resisted so far, and present a glimpse of an approach to study the problem grouping together Schur functions, combinatorics of words, and oriented matroids.