  Building sets were introduced in the study of wonderful compactifications of
hyperplane arrangement complements and were later generalized to finite
meet-semilattices. Convex geometries, the duals of antimatroids, offer a robust
combinatorial abstraction of convexity. Supersolvable convex geometries and
antimatroids appear in the study of poset closure operators, Coxeter groups,
and matroid activities. We prove that the building sets on a finite
meet-semilattice form a supersolvable convex geometry. As an application, we
demonstrate that building sets and nested set complexes respect certain
restrictions of finite meet-semilattices unifying and extending results of
several authors.
