# Minkowski sum alternative notation [closed]

The Minkowski sum of two polytopes is pretty neat. I'm working with a lot of them right now, and Minkowski sum of simplices, which write as stuff like:

\sum y_I \text{ conv } \{e_i | i \in I \}


So my problem arises when I want to talk about the vector space freely generated by these objects, so I have formal sums and Minkowski sums which should absolutely not have the same notation. When I'm adding two polytopes is all good as I can use '+' and '+^M' and is reasonably clear. For the sum symbol I tried to use the '^M' trick but looks bad (looks liek M is one of the limits of the sum).

Any ideas? Anyone that had a similar problem? Thanks!

EDIT: MWE as per requested:

\documentclass{article}
\usepackage[utf8]{inputenc}

\title{Minkowski Sum}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage[utf8]{inputenc}
\usepackage[english]{babel}

\begin{document}
Let be the Minkowski sum $Q_I = \sum_{I \subseteq J} \text{ conv } \{e_j | j \in J \}$ and consider the formal sum of polytopes:
$$\sum_{I \subseteq J \subseteq \{1, \cdots , n\}} \text{ conv } \{e_j | j \in J \} = \sum_{J \subseteq \{1, \cdots , n\}} 2^{\# J } \text{ conv } \{e_j | j \in J \}$$

Which is dangerously confusing as it is completely different from
$$\sum_{I \subseteq J \subseteq \{1, \cdots , n\}} \text{ conv } \{e_j | j \in J \}$$
\end{document}


## closed as off-topic by Romain Picot, Stefan Pinnow, TeXnician, Paul Gaborit, ThorstenJun 21 '17 at 11:37

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question does not fall within the scope of TeX, LaTeX or related typesetting systems as defined in the help center." – Romain Picot, Stefan Pinnow, TeXnician, Paul Gaborit, Thorsten
If this question can be reworded to fit the rules in the help center, please edit the question.

• Do you want us to advise you a good notation? If yes, then ask on math.stackexchange. If you want help typesetting something, then tell what you want to typeset. – Michael Fraiman Jun 21 '17 at 8:39
• And also, use \mathrm{conv}, not \text{conv}. – Michael Fraiman Jun 21 '17 at 8:39
• Could you just use some notation around a set to mean the formal vector space freely generated by it, and then the sums are all Minkowski sums, within that vector space or within the origin vector space? – Benjamin McKay Jun 21 '17 at 8:40
• @AndrewSwann is it possible to put the superscript before the sum symbol? – PenasRaul Jun 21 '17 at 9:31
• Yes, \sideset{^M}{}\sum..., see the amsmath documentation – Andrew Swann Jun 21 '17 at 10:44

In the comments, the OP mused whether a pre-superscript could be added to the \sum, as that might be sufficient to clarify the notation. The answer is, yes it can, but the actual syntax, in this particular case, turns out to be a little tedious. Normally ^2x_1 is sufficient for getting a superscript 2 before the x. However, in this case,

^M\sum_{...}


has the problem that if the underset is too wide, the M gets pushed leftward. At first, one might think that adding a group

{^M\sum}_{...}


might fix it, but that breaks the underscript, converting into a normal subscript. The normal fix for this is to add \limits to the scripts, but that can only be applied to math operators, as in

\mathop{^M\sum}\limits_{...}


The only problem with this is that it centers the underset not directly under the \sum, because of the width of the M. So the final correction is to lap the pre-script as in

\mathop{\mathllap{^M}\sum}\limits_{...}


The \mathllap requires the mathtools package. Thus, we finally arrive at

\documentclass{article}
\usepackage{amsmath,mathtools}
\usepackage{amssymb}
\begin{document}
$\mathop{\mathllap{^M}\sum}\limits_{I \subseteq J \subseteq \{1, \cdots , n\}} \text{ conv } \{e_j | j \in J \}$

$\mathop{^M\sum}\limits_{...}$
\end{document}


The amsmath package provides the \sideset command whose syntax is

\sideset{^a_b}{^c_d}\sum


placing a as presuperscript, b as a presubscript, and c, d in corresponding positions after. Thus you could use \sideset{^M}{}\sum for your situation:

\documentclass{article}

\usepackage{amsmath}

\DeclareMathOperator{\conv}{conv}

\begin{document}

\begin{equation*}
\sideset{^M}{}\sum_{I\subset J\subset\{1,\dots,n\}}
\conv\{e_j\mid j\in J\}
\end{equation*}

\end{document}


You can package this up in a new command \msum as follows:

\newcommand{\msum}{\sideset{^M}{}\sum}


Additionally, as Arzigoglu mentions, in the particular case with the long subscript, you might consider using \mathclap from the mathtools package to improve the spacing

\documentclass{article}

\usepackage{mathtools}

\DeclareMathOperator{\conv}{conv}
\newcommand{\msum}{\sideset{^M}{}\sum}

\begin{document}

\begin{equation*}
\msum_{\mathclap{I\subset J\subset\{1,\dots,n\}}} \mkern10mu
\conv\{e_j\mid j\in J\}
\end{equation*}

\end{document}


• +1. I prefer this solution. Probably something like \sideset{^M}{}\sum_{\mathclap{I\subset J\subset\{1,\dots,n\}}}\;\;\conv\{e_j\mid j\in J\} with mathtools looks better. – Arzigoglu Jun 21 '17 at 11:32
• @Arzigoglu Thank you. Yes, depending on the circumstances, using \mathclap from mathtools can indeed improve the spacing. – Andrew Swann Jun 21 '17 at 11:45
• Very instructive, as \sideset might be useful for other stuff as well. Thanks! – PenasRaul Jun 21 '17 at 12:43