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On inscribed trapezoids and affinely 3-regular maps
- Part of
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 154 / Issue 4 / August 2024
- Published online by Cambridge University Press:
- 12 May 2023, pp. 1024-1032
- Print publication:
- August 2024
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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.
