Drawing the region $A$ with inequality $|s_1| + |s_2| \leq 1$

Question

Currently I have trouble understanding this region $A = \{(s_1, s_2) \in \mathbb{R}^2 : |s_1| + |s_2| \leq 1\}$, since I do not really know how to transform the inequality $|s_1| + |s_2| \leq 1$. Is there a general receipt how to rearrange these kind of equalites to determine the bounds of $s_1$ and $s_2$?

Answer 1

Generally when you face an absolute value, try to break its domain to regions where it is definitely positive or definitely negative. For this case consider four cases

$$\begin{cases}s_1+s_2\le1&s_1\ge0,s_2\ge0 \\-s_1+s_2\le1 &s_1<0,s_2\ge0 \\ s_1-s_2\le1&s_1\ge0,s_2<0 \\-s_1-s_2\le1 &s_1<0,s_2<0\end{cases}$$

after ploting each of these regions, you get this diamond shape. It's actually $l_1$-norm ball is $\mathbb{R}^2$.

here is a more detailed figure that helps you figure out what is happening.