Shortest Paths and Convex Hulls in 2D Complexes with Non-Positive Curvature

Globally non-positively curved, or CAT(0), polyhedral complexes arise in a number of applications, including evolutionary biology and robotics. These spaces have unique shortest paths and are composed of Euclidean polyhedra, yet many properties of convex hulls in Euclidean space fail to transfer over. We give examples of some such properties. For 2-dimensional CAT(0) polyhedral complexes, we give polynomial-time algorithms for computing convex hulls using linear programming and for answering shortest path queries from a given source point.