Question 11
A circle with a circumscribed and an inscribed square centered at the origin $O$ of a rectangular coordinate system with positive $x$- and $y$-axes $OX$ and $OY$ is shown in each figure $\mathrm{I}$ to $\mathrm{IV}$ below.
The inequalities $$
\begin{aligned} |x|+|y| &\le \sqrt{2(x^2+y^2)}\\ &\le 2\operatorname{Max}(|x|,|y|). \end{aligned}
$$ are represented geometrically by the figure numbered
*Geometric representation:* An inequality $f(x,y)\le g(x,y)$ for all $x,$ $y$ is represented by a figure showing, for a typical real number $a,$ the containment $$
\begin{aligned} &\{(x,y):g(x,y)\le a\}\\ &\quad\subset \{(x,y):f(x,y)\le a\}. \end{aligned}
$$
The inequalities $$
\begin{aligned} |x|+|y| &\le \sqrt{2(x^2+y^2)}\\ &\le 2\operatorname{Max}(|x|,|y|). \end{aligned}
$$ are represented geometrically by the figure numbered
*Geometric representation:* An inequality $f(x,y)\le g(x,y)$ for all $x,$ $y$ is represented by a figure showing, for a typical real number $a,$ the containment $$
\begin{aligned} &\{(x,y):g(x,y)\le a\}\\ &\quad\subset \{(x,y):f(x,y)\le a\}. \end{aligned}
$$