What's the notation of a set with multiple indices?

Question

I want to take the maximum value of a set of values with three different indices:

$Y = \max (X_{1,1,1}, ..., X_{k,j,i})$

where: $i \in \{1,...,n\}$, $j \in \{1,...,m\}$, $k \in \{1,...,o\}$.

What would be the right notation for this?

Answer 1

$$Y = \max_{i, j, k}(X_{1,1,1}, \dots)$$ is unambiguous as long as you've defined the ranges of $i, j, k$. If you really want to be on the safe side you could put $1 \leq i \leq n$, $1 \leq j \leq m$, $1 \leq k \leq o$ below the $\max$.