On inscribed trapezoids and affinely 3-regular maps

We show that any embedding R-d ? R2d+2?(d)-1 inscribes a trapezoid or maps three points to a line, where 2(? (d)) is the smallest power of 2 satisfying 2(? (d))= ?(d), and ?(d) denotes the Hurwitz-Radon function. The proof is elementary and includes a novel application of nonsingular bilinear maps. As an application, we recover recent results on the nonexistence of affinely 3-regular maps, for infinitely many dimensions d , without resorting to sophisticated algebraic techniques.