On inscribed trapezoids and affinely 3-regular maps

TL;DR

This paper proves that certain high-dimensional embeddings necessarily contain inscribed trapezoids or map three points to a line, using elementary methods and nonsingular bilinear maps, and applies this to nonexistence results for affinely 3-regular maps.

Contribution

It introduces an elementary proof involving nonsingular bilinear maps for inscribed trapezoids and affinely 3-regular maps, avoiding complex algebraic techniques.

Findings

Any embedding from  to d+2^{\u03b3(d)}-1 inscribes a trapezoid or maps three points to a line.

The proof is elementary and employs nonsingular bilinear maps.

Results extend nonexistence of affinely 3-regular maps to infinitely many dimensions.

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 $2^{\gamma(d)} \geq \rho(d)$, 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.