# What to call a function that is negative on a set

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?

• Boundary defining functions are sort of like that. Mar 5, 2021 at 19:18

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 $$f$$.