Folks,
See attached pdf detailing some plots in $\displaystyle \mathbb{R}^2$. How were these graphs determined for 1-, and $\displaystyle \infty$- norms?
For 1-norm, the inequality is |x| + |y| <= 1. You can consider each of the four quadrants and expand |x| and |y| correspondingly. For example, in the second quadrant, x < 0 and y > 0, so |x| = -x and |y| = y. Therefore, the inequality is y - x <= 1, or y <= x + 1.
For $\displaystyle \infty$-norm, the inequality is max(|x|, |y|) <= 1, which is equivalent to |x| <= 1 and |y| <= 1, or -1 <= x, y <= 1.