On inscribed trapezoids and affinely 3-regular maps

Abstract

We show that any embedding ${\mathbb{R}}^d\to {\mathbb{R}}^{2d+2^{\gamma (d)}-1}$ inscribes a trapezoid or maps three points to a line, where $2^{\gamma (d)}$ is the smallest power of $2$ satisfying $2^{\gamma (d)}\ge \rho (d)$, and $\rho (d)$ 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.

8 pages; new version includes improved results and significant changes to exposition

References in corpus (1)

• Fibrations of ${\mathbb{R}}^3$ by oriented lines