Unit Balls in \(\mathbb{R}^2\)
For an intuition for how the metric used can effect the unit ball, consider the following sketches of closed unit ball centred at \((0, 0)\) in \(\mathbb{R}^{2}\) (\(\overline{B_{1}((0, 0))}\)) under different metrics.
Euclidean Metric
\[ d_2(x, y) = \sqrt{(x_{1} - y_{1})^{2} + (x_{2} - y_{2})^{2}}\]
Chebyshev Metric
\[ d(\boldsymbol{x}, \boldsymbol{y}) = \max\{|x_{1} - y_{1}|, |x_{2} - y_{2}|\}\]
Taxicab Metric
\[ d(\boldsymbol{x}, \boldsymbol{y}) = |x_{1} - y_{1}| + |x_{2} - y_{2}|\]
Discrete Metric
\[d(x, y) = \begin{cases}
0 & \text{if } x = y \\
1 & \text{if } x \neq y \\
\end{cases}\]
In case it is not clear, the closed unit ball here is the whole space.
This example is particularly subtle, because unlike the others where it is clear the difference when the ball is open, in this case the open unit ball under the discrete metric contains only the point \((0, 0)\).