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I have a pretty simple function that I want to express, but I'm having trouble coming up with a pleasing notation to express it. The function is:

$h(y) = \arg \min_x |f(x, y)|$

$g(y) = f(h(y), y)$

In English, "$g$ is the value of $f$ at which its absolute value is minimized". Is there a standard symbol for this sort of thing, or a more compact notational representation that I'm missing?

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You can use the infimum:

$$g(y) = \inf \{ |f(x,y)| : x \in D(y) \}$$

whereby $D(y)$ is the domain of $f(\cdot,y)$. In the case of $f: \mathbb R^2 \rightarrow \mathbb R$ you have:

$$g(y) = \inf \{ |f(x,y)| : x \in \mathbb R \}$$

or

$$g(y) = \inf_{x\in\mathbb R} |f(x,y)|$$

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