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On $\mathbb{R}^2$ we have a metric defined by $d(x,y)=|x_1- y_1|+ |x_2- y_2|$. Describe and illustrate $B_1(0,0)$, the ball of radius $1$ centered at the origin $(0,0)$.

SOLUTION By definition $B_1(0, 0)=\{(x,y)\in \mathbb{R}^2 : |x-0|+|y-0|<1\}$ Since the ball is at the origin and it has the radius one, then I find that it is the set of all points within the rectangle having the corner points $(0, 1)$, $(1, 0)$, $(0,-1)$, and $(-1,0)$ Is my idea correct? Thanks.

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