Lifting simplicial complexes to the boundary of convex polytopes

A simplicial complex CC on a dd-dimensional configuration of nn points is kk-regular if its faces are projected from the boundary complex of a polytope with dimension at most d+kd+k. Since CC is obviously (n−d−1)(n−d−1)-regular, the set of all integers kk for which CC is kk-regular is non-empty. The minimum δ(C)δ(C) of this set deserves attention because of its link with flip-graph connectivity. This paper introduces a characterization of δ(C)δ(C) derived from the theory of Gale transforms. Using this characterization, it is proven that δ(C)δ(C) is never greater than n−d−2n−d−2. Several new results on flip-graph connectivity follow. In particular, it is shown that connectedness does not always hold for the subgraph induced by 33-regular triangulations in the flip-graph of a point configuration.