No CrossRef data available.
Article contents
On inscribed trapezoids and affinely 3-regular maps
Published online by Cambridge University Press: 12 May 2023
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.
MSC classification
- Type
- Research Article
- Information
- Copyright
- Copyright © The Author(s), 2023. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh
References
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230511141123455-0888:S0308210523000446:S0308210523000446_inline314.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230511141123455-0888:S0308210523000446:S0308210523000446_inline315.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230511141123455-0888:S0308210523000446:S0308210523000446_inline316.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230511141123455-0888:S0308210523000446:S0308210523000446_inline317.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230511141123455-0888:S0308210523000446:S0308210523000446_inline318.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230511141123455-0888:S0308210523000446:S0308210523000446_inline319.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230511141123455-0888:S0308210523000446:S0308210523000446_inline320.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230511141123455-0888:S0308210523000446:S0308210523000446_inline321.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230511141123455-0888:S0308210523000446:S0308210523000446_inline322.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230511141123455-0888:S0308210523000446:S0308210523000446_inline323.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230511141123455-0888:S0308210523000446:S0308210523000446_inline324.png?pub-status=live)