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$$

d_1(\mathbf{x},\mathbf{y}) = \max\limits_{1\leq i\leq n}|x_i - y_i|

$$

Prove that the ball $B(\mathbf{a},r)$ has the geometric appearance indicated:

In $(\mathbb{R}^2,d_1)$, a square with sides parallel to the coordinate axes.

Consider the ball $B((x_1,y_1),r)$ where $r > 0$.

Let $(x_2,y_2)\in B((x_1,y_1),r)$.

We know that $|x_1 - x_2| < r\iff -r < x_1 - x_2 < r$ and $|y_1 - y_2| < r\iff -r < y_1 - y_2 < r$ by construction and which forms a square with sides parallel to the coordinate axes.