In $\mathbb{R}^2$, the Taxicab metric is defined by $d(x,y) = |x_1 – y_1| + |x_2 – y_2|.$
In this metric, describe/draw the open unit ball, closed unit ball, and the unit sphere which are based at the origin with radius $1$.
Here is my work:
Since it says this problem based at the origin,
Open unit ball is $B(0,1) = \{x \in M : d(x,0) < 1\}$ which means open ball with center origin and radius $1$.
Closed unit ball is $B[0,1] = \{x \in M : d(x,0) <= 1\}$
Unit sphere is $S(0,1) = \{x\in M : d(x,0) = 1\}$
But I'm not sure how to draw them. I draw a dotted circle for the open ball with center $(0,0)\,$ and $r = 1$. But I doubt because I draw within negative quadrant. Like just a circle with $(0,0)\,$ and $r = 1$. I hope you understand my description….
Best Answer
Do you know what the taxicab metric is? Find its definition and try to find lots of points in $\mathbb R^2$ that will have distance precisely $1$ from the origin. That will be the boundary of $B(0,1)$, and it will not look anything like a circle.