$26 \cdot 8$ A circle with an inscribed square and a circumscribed square has its center at the origin $O$ of a rectangular coordinate system with $x$-axis $O x$ and $y$-axis $O y$, as shown in the figures $\mathrm{I}$ to $\mathrm{N}$ below. The geometric meaning of the inequality $|x|+|y| \leqslant \sqrt{2\left(x^{2}+y^{2}\right)} \leqslant 2 \max (|x|,|y|)$ is represented by the figure
(A) I.
(B) $I I$.
(C) III.
(D) IV.
(E) None of the above.
(24th American High School Mathematics Examination, 1973)
[Solution] For some positive integer $r$, observe the right figure.
The equation of the outer square is $|x|+|y|=r$, and the points on or inside the square satisfy:
(1) $|x|+|y| \leqslant r$.
The equation of the circle is $x^{2}+y^{2}=\frac{r^{2}}{2}$, so the points on the circle satisfy:
(2) $\sqrt{2\left(x^{2}+y^{2}\right)}=r$.
The equation of the inner square is $\max (|x|,|y|)=\frac{r}{2}$, so the points on or outside the square satisfy:
(3) $r \leqslant 2 \max (|x|,|y|)$.
Thus, from (1), (2), and (3), we can see that the points on the circle satisfy:
$$
|x|+|y| \leqslant r=\sqrt{2\left(x^{2}+y^{2}\right)} \leqslant 2 \max (|x|,|y|) \text {. }
$$

Therefore, the answer is $(B)$.