On stars and links of shellable polytopal complexes

The vertex stars of shellable polytopal complexes are shown to be shellable. The link of a vertex v of a shellable polytopal complex is also shown to be shellable, provided that all facets of the star of v are simple polytopes, or (more generally) if there exists a shelling F 1 , … , F n of the star of v such that, for every 1 < j ⩽ n , the intersection of F j with the previous facets is an initial segment of a line shelling of the boundary complex of F j .