arg min and arg max

For a real-valued function $f$ with domain $S$, $\arg {\min }_{x\in S}f(x)$ is the set of elements in $S$ that achieve the global minimum in $S$,

$$\arg \underset{x\in S}{\min }f(x)=\{x\in S:f(x)=\underset{y\in S}{\min }f(y)\}.$$

$\arg {\max }_{x\in S}f(x)$ is the set of elements in $S$ that achieve the global maximum in $S$,

$$\arg \underset{x\in S}{\max }f(x)=\{x\in S:f(x)=\underset{y\in S}{\max }f(y)\}.$$