Minimax and Maximin of Functions of Two Variables

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The function $f(x,y)$ is continuous in the rectangle

a \leq x \leq a', \qquad b \leq y \leq b' .

Then its maximum for a given $x$ and $b \leq y \leq b'$ (along a segment parallel to the $y$ -axis)

\max _{y} f(x,y) = \varphi (x)

is a continuous function of $x$ .

We interpret the surface

z = f(x,y)

in a rectangular coordinate system $x,y,z$ with vertical $z$ -axis as a topographical surface in a mountainous region. Then the curve

z = \varphi (x)

(in the $x,z$ -plane) is the skyline of the range (as it appears when seen from a faraway point of the $y$ -axis) and the minimax

\min _{x} \varphi (x) = \min _{x} \max _{y} f(x,y)

(the minimum of the maxima) refers to the lowest point in the skyline.

(continued). Show that

\max _{y} \min _{x} f(x,y) \leq \min _{x} \max _{y} f(x,y) .

(continued). Which one of the two signs, $<$ and $=$ , is valid in the example

f(x,y) = 1 - (x-y+1)^{2},

$a=0$ , $a'=2$ , $b=0$ , $b'=4$ ?

Add to the assumptions of Problem 223 that $f(x,y) > 0$ . Then

\lim _{n \to \infty } \left[\int _{a}^{a'} \left(\int _{b}^{b'} \left[f(x,y)\right]^{n} \, dy\right)^{-1} dx\right]^{-\frac{1}{n}} = \min _{x} \max _{y} f(x,y) .

References

[PóSz98] Pólya G., Szegő G. Problems and Theorems in Analysis. Springer, 1998.