On regular maps and parallel lines

TL;DR

This paper improves a recent mathematical result by showing the existence of four points in Euclidean space with specific properties such that the images of certain pairs under a continuous map are parallel, revealing new geometric insights.

Contribution

The paper advances previous work by establishing more general conditions for the existence of parallel vectors in the images of four points under a continuous map.

Findings

Existence of four points with parallel difference vectors under continuous maps.

Refinement of conditions compared to prior results by Frick and Harrison.

Improved understanding of geometric configurations in Euclidean spaces.

Abstract

Let $f: R^{m+1}\to R^{m+2^r}$, where $2^{r-1}\leq m+1 <2^r$, be a continuous map. Improving a recent result of Frick and Harrison, we show that there are $4$ points $x_0,\, x_1,\, y_0,\, y_1$ in $R^m$, which are distinct if $m+1\not=2^{r-1}$, and satisfy $x_0\not=x_1$, $y_0\not=y_1$, $\{ x_0, x_1\} \not=\{ y_0,y_1\}$ if $m+1=2^{r-1}$, such that the vectors $f(x_1)-f(x_0)$ and $f(y_1)-f(y_0)$ are parallel.