Editor asks me to define arg lim min

Question

I have recently submitted a paper to IEEE Trans. on Information Theory. I got the reviewer comments and one reviewer seemed to me like he/she was obsessive with the notations. For example he was unhappy that I used $\mathbb{R}_{\geq 0}$ instead of $\mathbb{R}_{+}$ for positive real numbers including zero.

In one of his/her comments he asked me to define $$\arg \lim \min$$ which I used in my paper. Before this one, I had used $\lim$ and $\min$ before. I am totally lost about this comments. These are in his list of major comments which led to the rejection of the paper although he/she thought that the paper is a good one in content.

I am now confused about what to do and/or how ro react to this comment. For example saying something like this: "where arg stands for the argument of bla bla bla" doesnt seem to me meaningful because if this is the case I also have to define $\min$ as "where min stand for the minimum of.."

What should be the proper way to react? Is there something that I am missing about this comment?

Answer 1

It might be worth you (cross-)posting this to the Academia StackExchange since this is, in many ways, more about the presentation of your paper than mathematics. However, especially in mathematics, good notation can make ideas much clearer while bad notation can make things harder to understand. Your reviewer may seem to be making a lot of fuss about notation, but they presumably believe that you're making it harder for your readers to follow your argument with your current choice.

Whether that's true or not is not something we can advise on, as we cannot see the paper ourselves.

I personally have no idea what you mean by $\mathrm{arg\ lim\ min}$. I know what the individual terms means, and I know what $\arg\min$ is, but that $\lim$ in the middle is both new and puzzling. To address this, you need to state what you mean by it, not by the individual terms in the phrase. For example,

$$\arg\min_{x \in D} f(x) := \{x \mid f(x)\leq g(y)\ \forall y\in D \}$$ defines $\arg\min$, so if (and I'm guessing here) $\arg\lim\min$ is a limiting version of that, perhaps

$$\arg\lim\min_{x \in D} f(x) := \{x \mid \lim_{n\rightarrow \infty}f_n(x)\leq g(y)\ \forall y\in D \}$$

is what is needed.

Answer 2

1. Write $[0, \infty)$ for positive real numbers including zero.

2. $\arg \lim \min$ is not a common notation. Give a definition and the reviewer will be satisfied.