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This is a soft question, hoping that it is still appropriate for this forum.
I need to describe twice the following region of $\mathbb{R}^k$ (i.e., we are in a $k$-dimensional Euclidean space, where $k \in \mathbb{N}-\{0,1,2\}$).
In detail, I have to state (in the abstract) that a constraint of a given optimization problem is the AABB defined as $[0,4-\sqrt{3}] \times [0,4-\sqrt{3}] \times [0,2] \times \cdots \times [0,2]$, where the closed interval $[0,2]$ appears $k-2$ times, while in the body of the article I would like to state the same product as above, by underlying the following construction: $$[0,2] \times \left[0,4-\sqrt{3}\right] \times \left[0,4-\sqrt{3}\right] \times \mathop{\Large\times}_{i=1}^{k-3}[0,2]$$ (there, I will replace the $\mathop{\Large\times}$ symbol with "\varprod").

Now, I am not sure if it would be fine to omit the last Cartesian product, I mean to simply write $[0,2] \times \left[0,4-\sqrt{3}\right] \times \left[0,4-\sqrt{3}\right] \mathop{\Large\times}_{i=1}^{k-3}[0,2]$. Otherwise, my last chance would be to forget about the construction and simply state the given box as $\left(\mathop{\Large\times}_{i=1}^{2} \left[0,4-\sqrt{3}\right] \right) \times \left( \mathop{\Large\times}_{j=3}^{k}[0,2]\right)$.

Thanks in advance for the kind suggestions.

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    $\begingroup$ For what it's worth, I like the convention that if $\boxtimes$ is some kind of multiplication operator, then the $\boxtimes$-product $\bigboxtimes_{i=1}^n X$ of $n$ copies of the same object $X$ is written $X^{\boxtimes n}$. So for example $E^{\otimes 2}$ for the tensor square, and so on. So if you want to insist that you're taking the $n$-fold Cartesian product of a set, you can write it $X^{\times n}$. $\endgroup$
    – Gro-Tsen
    Commented Jan 30 at 23:42
  • $\begingroup$ (re my previous comment: apparently MathJax doesn't have \bigboxtimes, but I think you can guess what you should read) $\endgroup$
    – Gro-Tsen
    Commented Jan 30 at 23:43
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    $\begingroup$ Thank you @Gro-Tsen! I've just taken another look at the definition and I finally agree that knowing the set notation $X^n:= X \times \cdots \times X$ (where $X$ appears exactly $n$ times) can be taken as a minimum requirement for any reader of a graph theory article. Then, if I use $\left[0,4-\sqrt{3}\right] \times \left[0,4-\sqrt{3}\right] \times [0, 2]^{k-2}$ in the abstract, it is more than reasonable to be allowed to write also $[0,2] \times \left[0,4-\sqrt{3}\right] \times \left[0,4-\sqrt{3}\right] \times [0, 2]^{k-3}$ in Section 2. $\endgroup$
    – Marco Ripà
    Commented Jan 31 at 15:18
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    $\begingroup$ A meta post related to this question: meta.mathoverflow.net/questions/5862/a-mathjax-latex-question $\endgroup$
    – Emil Jeřábek
    Commented Feb 1 at 8:58

1 Answer 1

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Certainly not the syntactically incorrect $[0,2] \times \left[0,4-\sqrt{3}\right] \times \left[0,4-\sqrt{3}\right] \mathop{\Large\times}_{i=1}^{k-3}[0,2]$.

Why not $[0,2]\times[0,4-\sqrt3\,]\times[0,4-\sqrt3\,]\times[0,2]^{k-3}$?

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  • $\begingroup$ It seems nice to look at... my only concern is about writing $[0,2]^{k-3}$ to indicate a cartesian product (I hope that the casual reader does not think of the most common $\prod_{j=3}^{k}$ or even dot product). If there are no risks, this would be a good solution. $\endgroup$
    – Marco Ripà
    Commented Jan 30 at 22:44
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    $\begingroup$ P.S. In the "old" abstract, I gave it as $[0,4-\sqrt{3}] \times [0,4-\sqrt{3}] \times [0,2] \times \cdots \times [0,2]$, maybe just adding a long horizontal curly bracket below $[0,2] \times \cdots \times [0,2]$ with "$(k-2)$-times" as its label would be nice for the abstract. No idea if this can be easier to read the first time. $\endgroup$
    – Marco Ripà
    Commented Jan 30 at 22:50
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    $\begingroup$ @MarcoRipà : For a set $S$, the standard way to read $S^m$ is as the $m$-fold cartesian product of $S$. Other ways to read this would be unusual and requiring explicit definitions. Also, not everything has to be completely defined in an abstract. Also, I don't think that an underlining curly bracket would be appropriate for an abstract, where heavy notations should be avoided. $\endgroup$
    – Iosif Pinelis
    Commented Jan 30 at 23:00
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    $\begingroup$ @Marco I agree with Iosif, but if you are super worried, then blah$\times [0,2]^{\times (k-2)}$ is another option. $\endgroup$
    – David Roberts ♦
    Commented Jan 31 at 10:43
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    $\begingroup$ @MarcoRipà : In addition, I do not remember ever seeing elsewhere using the nonexistent natively in TeX operator $\mathop{\Large\times}$ to write something like $\mathop{\Large\times}_{i\in I}S_i$ to denote the cartesian product of sets $S_i$ over $i\in I$. Also, I do not remember ever seeing something like $\prod_{i\in I}S_i$ denoting anything except the cartesian product of sets $S_i$ over $i\in I$. As for the dot product, it is only a binary operation and certainly requires a particular symbol to denote it. So, I don't understand your worry about $[0,2]^{k-3}$, especially in the abstract. $\endgroup$
    – Iosif Pinelis
    Commented Jan 31 at 11:02

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