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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2021
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2109.05985 |
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Table of Contents:
- We show that any embedding $\mathbb{R}^d \to \mathbb{R}^{2d+2^{γ(d)}-1}$ inscribes a trapezoid or maps three points to a line, where $2^{γ(d)}$ is the smallest power of $2$ satisfying $2^{γ(d)} \geq ρ(d)$, and $ρ(d)$ 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.