Lifting simplicial complexes to the boundary of convex polytopes

A simplicial complex C on a d -dimensional configuration of n points is k -regular if its faces are projected from the boundary complex of a polytope with dimension at most d + k . Since C is obviously ( n − d − 1 ) -regular, the set of all integers k for which C is k -regular is non-empty. The minimum δ ( C ) of this set deserves attention because of its link with flip-graph connectivity. This paper introduces a characterization of δ ( C ) derived from the theory of Gale transforms. Using this characterization, it is proven that δ ( C ) is never greater than n − 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 3 -regular triangulations in the flip-graph of a point configuration.