# Why is the norm ball a square in $\,\mathbb R^2\,$ under $\,l^\infty\,$ norm?

Suppose $\,x = \left(x_1, x_2\right)$, then $\,l^2\,$ norm ball is $\,\left\lbrace x\;\big\vert\;\, \sqrt{\left\lvert x_1 \right\rvert^2 + \left\lvert x_2 \right\rvert^2} \leq 1\right\rbrace$

Easily we can see that $\left\lvert x_2 \right\rvert = \pm\sqrt{1 - \left\lvert x_1 \right\rvert^2}$

And the set of $\,\left(x_1,x_2\right)\,$ forms a circle in $\, \mathbb R^2$.

Now suppose $\,x = \left(x_1, x_2\right)$, then $\,l^\infty\,$ norm ball is $\,\big\lbrace x\, \mid\,\left\|x\right\|_\infty \leq 1\big\rbrace$

where $\,\left\|x\right\|_\infty \leq 1 = \max\big\lbrace\left\lvert x_1 \right\rvert, \left\lvert x_2 \right\rvert\big\rbrace \leq 1,\,$ but in this case we cannot use the same technique as in $\,l^2\,$ case to write $\,x_2\,$ as a function of $\,x_1$.

Can anyone tell me how they came up with the following diagram?

• The $\ell^{\infty}$ norm of a point in $\mathbb{R}^2$ is just the maximum of the two coordinates, and the square encloses all points where neither of the two coordinates exceeds 1, right? Sep 14, 2015 at 17:18
• It's the limit $||x||_\infty = \lim \limits_{n \to \infty} ||x||_n$! Draw pictures as $n$ grows. Sep 14, 2015 at 17:22
• @gogurt How do you justify the line on the negative $x_1$ $x_2$ axis?, The vertical line on the left is clearly $|x_1| = -1$ and the horizontal line at the bottom is $|x_2| = -1$ which is both impossible Sep 14, 2015 at 17:22
• Think of this as $\left \{ (x_{1},x_{2}) \in \mathbb{R}|\left | x_{1} \right | \leq \left | x_{2} \right | \leq 1) \right \} \bigcup \left \{ (x_{1},x_{2}) \in \mathbb{R}|\left | x_{2} \right | < \left | x_{1} \right | \leq 1) \right \}$ Sep 14, 2015 at 17:23
• @IllegalImmigrant how are they impossible? Sep 14, 2015 at 17:24

Think of this as $$\left \{ (x_{1},x_{2}) \in \mathbb{R}|\left | x_{1} \right | \leq \left | x_{2} \right | \leq 1) \right \} \bigcup \left \{ (x_{1},x_{2}) \in \mathbb{R}|\left | x_{2} \right | < \left | x_{1} \right | \leq 1) \right \}$$
• Okay, so the trick is that $|x_1| \leq 1 \Leftrightarrow -|x_1| \geq -1$ Thanks that really was my confusion as to how you got the negative areas Sep 14, 2015 at 20:53
They looked for all points $(x,y)$ in the plane such that $|x|\leq 1$ and also $|y|\leq 1$.