Extending shelling orders and a hierarchy of functions of unimodal simple polytopes (Q1894364)

Suppose that, for some \(k < n\), there is an ordering of \(k\) facets of a simplicial \(d\)-polytope \(P\) with \(n\) facets such that the intersection of each of these facets with the union of all facets, that precede it in the ordering, is a nonempty union of \((d - 2)\)-faces. It was asked by H. Tverberg and recorded by G. Danaraj and V. Klee whether this initial ``segment'' can be extended to a shelling of all the facets. This question is open even for the duals of \(d\)-cubes, and so it has relations to the hierarchies of completely unimodal pseudo-Boolean functions (mapping the vertices of the \(d\)-cube into the reals). The author compares and combines the hierarchies and gives an extension to general simple polytopes.