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On inscribed trapezoids and affinely 3-regular maps

Part of: Classical differential geometry Differential topology

Published online by Cambridge University Press:  12 May 2023

Florian Frick  and
Michael Harrison Open the ORCID record for Michael Harrison [Opens in a new window]
Florian Frick
Affiliation:
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, United States Institute of Mathematics, Freie Universität Berlin, Arnimallee 2, 14195 Berlin, Germany (frick@cmu.edu)
Michael Harrison
Affiliation:
Institute for Advanced Study, 1 Einstein Drive, Princeton, New Jersey 08540, United States (mah5044@gmail.com)
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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.

Keywords

inscribed trapezoidsk-regular mapsnonsingular bilinear mapsHurwitz–Radon function

MSC classification

Primary: 53A07: Higher-dimensional and -codimensional surfaces in Euclidean $n$-space
Secondary: 57R40: Embeddings
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 154 , Issue 4 , August 2024 , pp. 1024 - 1032
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Copyright
Copyright © The Author(s), 2023. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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