Unfamiliar notation for min

Question

I'm reading a paper and came across some notation for a minimum that I have not seen.

Given $k \in \{ 1 .. n \}$ and a function that defines $j_k$ earlier in the text,

I see this notation: $$ \min_k (j_k) $$

What would the subscript on min indicate?

Answer 1

$\min_k$ means "among all possible / suitable $k$, the minimum value of ..."

Whatever comes after $\min_k$ is put into the "..." in the above sentence.

So, $\min_k j_k$ means "among all $k$, the minimum value of $j_k$"

In general, there are many notations using this paradigm of $$\operatorname{Operation}_{\text{index}}\text{expression}$$(where $\text{expression}$ uses $\text{index}$ in some way) to mean "take all the different values of $\text{expression}$ for all suitable $\text{index}$, and use $\operatorname{Operation}$ on them. Examples include

• $\bigcup_k I_k$ meaning "take all the different sets $I_k$, and use $\cup$ on them (take their union)
• $\sum_kf(k)$ meaning "take all the different values $f(k)$, and use $\sum$ on them" (for some reason we use $\sum$, instead of a big $+$).

and so on, for different operations like $\max, \bigcap, \prod, \bigoplus$ and $\bigwedge$. One might even argue that $\lim$ works like this, but that would probably be somewhat of a stretch.

Answer 2

The notation is read "min over k of..." meaning that you are free to adjust the value of $k$ in order to minimize the value of $j_k$. A clearer example would be:

$$\min_k A(m,k)$$

which means you're only free to vary $k$ (not $m$) in your search.

Clear?

Answer 3

In addition to the other answers, you can opt to expand this notation into the $\min$ function's n-ary argumentative form:

Given $k \in \{1,...,n\}$, we can express $$\min_k (j_k)$$ as $$\min(j_1, ..., j_n)$$