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When does $\min_x \max_yf(x,y) = \min_y \max_x f(x,y)$ hold for a real function $f(x,y)$?

Let $f(x,y)$ be a real function of the variables $x,y$ (which can be real vectors). Under what conditions do we have the following equality: $$\min_x \max_yf(x,y) = \min_y \max_x f(x,y)$$ For example, this equality is true if $f(x,y) = xy$ and $x,y$ are real scalars. Note that this is not the same as Von Neumann’s […]

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$\min_x \max_yf(x,y) = \min_y \max_x f(x,y)$

Let $f(x,y)$ be a real function of the variables $x,y$ (which can be real vectors). Under what conditions do we have the following equality: $$\min_x \max_yf(x,y) = \min_y \max_x f(x,y)$$ For example, this equality is true if $f(x,y) = xy$ and $x,y$ are real scalars. Note that this is not the same as Von Neumann’s […]

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Conditions for $\min_x \max_yf(x,y) = \min_y \max_x f(x,y)$?

Let $f(x,y)$ be a real function of the variables $x,y$ (which can be real vectors). Under what conditions do we have the following equality: $$\min_x \max_yf(x,y) = \min_y \max_x f(x,y)$$ For example, this equality is true if $f(x,y) = xy$ and $x,y$ are real scalars. Note that this is not the same as Von Neumann’s […]

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Ask Mathematics

Conditions for $\min_x \max_yf(x,y) = \min_y \max_x f(x,y)$?

Let $f(x,y)$ be a real function of the variables $x,y$ (which can be real vectors). Under what conditions do we have the following equality: $$\min_x \max_yf(x,y) = \min_y \max_x f(x,y)$$ For example, this equality is true if $f(x,y) = xy$ and $x,y$ are real scalars. Note that this is not the same as Von Neumann’s […]