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Let $Y$ be a nonempty region in $\mathbb{R}^n$. I am designing an algorithm which given a point $x_0$ outside $Y$ in a finite number of steps lead to a point $x_n∈ Y$. The way I do it is that I have a smooth function $f(x)$ which has the property that it is negative in $Y$ and positive outside $Y$ and it has only one local minimum, which is of course inside $Y$. So I get from $x_0$ towards $Y$ by minimizing the function $f(x)$.

Is there standard way to call such a function $f(x)$? This function "indicates" where $X$ is so I might want to call it "indicator function", but of course this is already taken as it denotes the characteristic function of $X$. Is there any other standard term?

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    $\begingroup$ Boundary defining functions are sort of like that. $\endgroup$
    – Ben McKay
    Mar 5, 2021 at 19:18

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In the context of Level-set method, such a function is called a level set function. The reason is that the boundary $\partial Y$ corresponds to the zero-level set of the function $f$.

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