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.