Any way to mathematically express the set of all argmin(f(x))?

Question

my question is: I would like to express in a mathematical way the first $argmin(f(x))$. The function $argmin$ returns the argument for the global minima, but, there is any way to express the set of all arguments that produce local minimums?

I would like to write an expression like $first argmin = min(argSmin(f(x)))$.

I hope my question is clear...

Thanks for your help :)

Answer 1

Your question is not very clear, but I can answer "is any way to express the set of all arguments that produce local minimums?":

Let $f : X \to Y$. The set of local minimizers is: $$\{ x \in X : \exists \varepsilon > 0 : f(x) \leq f(z) \; \forall z \in X, ||z-x|| \leq \varepsilon \}.$$

Answer 2

My notation is also that $\arg \min$ returns a set. If you chose differently you can write: $firstmin_D f=\min\lbrace x\in D, x=\min_D f\rbrace$