Extending shelling orders and a hierarchy of functions of unimodal simple polytopes

Let P be a simplicial d-polytope with n facets ((d − 1)-dimensional faces) in R^d. A shelling of P is an ordering of the facets of P such that the intersection of each facet F with the union of all facets that precede it the ordering is a nonempty union of (d − 2)-faces of F. The following open question was raised by Tverberg and is recorded in 4]. Suppose for some k < n, there is an ordering of k of the facets of P so that the intersection of each of these facets with the union of all of the facets that precede it in the ordering is a nonempty union of (d − 2)-faces. Can this initial “segment” be extended to a shelling of all the facets? This question is open even in the case that P is the dual of the d-dimensional hypercube. The question in this case has resurfaced several times since G. Danaraj and V. Klee (1978) in a variety of forms. It is related to the hierarchies of completely unimodal pseudo-Boolean functions studied in P.L. Hammer et al. (1988), the author (1988) and D. Wiedemann (1986). (A pseudo-Boolean function is a function mapping the vertices of the d-dimensional hypercube into the reals). In this paper, the hierarchies are compared and combined. This hierarchy is then extended to general simple polytopes, and the relationship to the above open question is explained.