Central and parallel projections of polytopes (Q1084675)

Let X and Z denote two point sets which are in one-one correspondence. Subsets of X are denoted by X(I) and the corresponding subset of Z by Z(I). For a set W, let pW denote the set of all linear combinations of points in W with either all nonnegative or all nonpositive coefficients. The following theorem is proved using the technique of Gale diagrams: The set Y in \(R^{k-1}\) is a projection of the vertex set X of the k- polytope P iff there is a point common to the relative interiors of all the sets pZ(I), such that X(I) is a cofacet of P, Z denoting a Gale transform of Y.