Give an example of two different metrics on $\mathbb{R}^2$ whose restrictions to the $x$-axis are equal.
Consider the two metrics $d_2$ and $d_1$ on $\mathbb{R}^2$ where $d_2$ denotes the usual Euclidean distance and $d_1$ denotes the "taxicab" metric on $\mathbb{R}^2$. Namely, $d_1(x,y)=|x_2-x_1|+|y_2-y_1|$.
Restricting $d_2$ to the $x$-axis gives us,
$$d_2((x_1,0),(x_2,0))=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}=\sqrt{(x_2-x_1)^2+(0-0)^2}=\sqrt{(x_2-x_1)^2}=|x_2-x_1|$$
Similarly, restricting $d_1$ to the $x$-axis yields,
$$d_1((x_1,0),(x_2,0))=|x_2-x_1|+|y_2-y_1|=|x_2-x_1|+0=|x_2-x_1|$$
Hence, $d_2$ and $d_1$ are equivalent when restricted to the $x$-axis.
Is this correct?
Best Answer
Your proof is correct. Additionally, note that the same proof shows that all metrics induced by $L^p$ norms are equal when restricted to $x$-axis for all finite $p$.