Lawrence Oriented Matroids and a Problem of McMullen on Projective Equivalences of Polytopes

Consider the following question introduced by McMullen: Determine the largest integern = f(d) such that any set of n points in general position in the affine d -space Rdcan be mapped by a projective transformation onto the vertices of a convex polytope. It is known that 2 d + 1 ≤ f(d) < (d + 1)(d + 2) / 2 and it is conjectured that f(d) = 2 d + 1. In this paper, we show that f(d) < 2 d + ⌈d + 1 2 ⌉ by using a well-known oriented matroid generalization of the above problem.