On inscribed trapezoids and affinely 3-regular maps

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