On affinely 3-regular maps and trapezoids

We show that any embedding $mathbb{R}^d to mathbb{R}^{2d+rho(d)-1}$ inscribes a trapezoid or maps three points to a line, where $rho(d)$ denotes the Radon-Hurwitz function. This strengthens earlier results on the nonexistence of affinely $3$-regular maps for infinitely many dimensions $d$ by further constraining four coplanar points to be the vertices of a trapezoid.