论文标题
在常规地图和平行线上
On regular maps and parallel lines
论文作者
论文摘要
令$ f：r^{m+1} \ to r^{m+2^r} $，其中$ 2^{r-1} \ leq m+1 <2^r $，是连续的地图。改善了弗里克和哈里森的最新结果，我们表明有$ 4 $ x_0，\，\，x_1，\，\，y_0，\，\，y_1 $ in $ r^m $中的$ r^m $，如果$ m+m+1 \ not = 2^{r-1} $，并且满足$ x_0 \ x_1 $ x_1 $ y_1 $，$ y_1 $， x_1 \} \ not = \ {y_1，y_1 \} $如果$ m+1 = 2^{r-1} $，以使vectors $ f（x_1）-f（x_0）$和$ f（y_1）-f（y_1）-f（y__0）$是平行的。
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.
