Various typos/comments

[julian-upc]

I'm going to put these into an issue, because the commits are now too sprinkled with comments for anyone to make sense of them.

1. Please use the unstarred versions of definition/theorem/etc via the code

\numberwithin{equation}{section}

\newtheorem{theorem}{Theorem}
\numberwithin{theorem}{section}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
...

2. Please include myclass.cls into the source tree because else the code doesn't compile.

[julian-upc]

Section 1:

Definition of Matroid: The command \choose is actually TeX, not LaTeX, and doesn't work as expected. Use \binom{[n]}{r} instead. You also need to reword the definition so that you don't use B before it is defined.

Also, please don't use shorthand notation like \exists, because that makes the formulas too compressed to read quickly.

Remember that your reader has a lot of other things to do, so anything that you can do for her to be able to absorb your content faster is a big bonus! And it really helps to use text instead of formulae.

[julian-upc]

Section 2.1:

• define S_n
• put the (very very cool!!!) picture into a

\begin{figure}
...
\caption{...}
\label{fig:permutahedron}
\end{figure}

environment, and refer to it as "as shown in Figure~\ref{fig:permutahedron}" instead of "as shown in the picture"

• "Flags could" -> "Flags can also be" (don't use conditional when you can use active verbs).
• "One example of how the three points of view interact is the following:"

\begin{center}
\begin{tabular}{ccc}
Flag 
& 
ordered partition 
& 
face of permutahedron 
\\\hline
${{3},...$
&
$3|124$
&
...
\end{tabular}
\end{center}

[julian-upc]

Section 2.2:

Def hypersimplex: It's not enough to just write a formula, you actually need to write a complete sentence. Remember that your job is to make your reader's experience as pleasant as possible, and by the law of conservation of difficulty this means that you will have to put in a lot of work for her experience to be easy!
Also, use \cdots inside the sum.

Def Generalized PH: same here, you need a complete sentence

"Came" is the wrong tense. Also, your sentence isn't true: the vertices of a GP don't have to have 0/1 coordinates, they can be anywhere

Def Matroid: Don't capitalize matroid in the body of the definition.

[julian-upc]

3. Coxeter groups:

• I already made some comments in the corresponding commits.
• "It is not needed to label it" -> "usually the labels are omitted".
• In the picture, make the first edge solid and the second edge dotted.
• In your displays, use \text{ if } (note the spaces) to make the "if" not render as math

[julian-upc]

Section 5.1:

• The definition of M-convex needs to go after the definition of regular subdivision.
• Def 3TPR:
  • comma after \notin S$
  • In the display, use \min\Big\{ ... \Big\}, and put spaces after the commas, like this: ,\;
  • You should make explicit that you are identifying subsets of [n] with vertices of Delta(k,n)
  • you write S_ij when you mean S\cup\{i,j\}. You can abbreviate this to Sij if you like, but S_{ij} is misleading (as well as undefined).
• Theorem: "hold", not "holds", because 3TPR is plural
• Proof:
  • h_ij also depends on S, but that isn't apparent from your notation
  • What is a "regular division polytope"?
  • The display confuses me: A is a subset of M; do you mean a basis of M? Who is P? what does A run over to achieve the minimum?
  • indexes -> indices
  • allways -> always
  • what "terms" are we removing?
  • It's not good form to say something is "trivial". All that this means is that you understood something in the moment you wrote it down, but just a few months onwards you might not understand it anymore. Also, it can make your readers feel bad if they don't immediately understand the reason for your statement, and your number one priority is to make their experience reading your paper to be enjoyable, easy and fun!
  • So instead of saying something is trivial, just say why it's true, and everyone is happy.
  • Without understanding Q I can't tell whether "wlog" Sik, Sjl \in Q. Maybe you could also say explicitly why we can assume that?
  • Something is wrong with the spacing in the numbering of your equations (1) and (2)
  • Under de -> under the
  • (1) and (2) imply (there are two of them)

[julian-upc]

Section 5.2:

• Definition: Use \conv, and remember to write complete and varied sentences.
• One way in which you could describe your construction is to divide the 2n dimensions into n "up" and n "down",

| up dimensions   |  1 |  2 |  3 | ... | n  |
|-----------------+----+----+----+-----+----|
| down dimensions | -1 | -2 | -3 | ... | -n |

and saying that a vertex (v_1,...,v_n) of the cube gets assigned a sequence of vectors

| 1 |
|---|
| 0 |

or

| 0 |
|---|
| 1 |

depending on whether v_i=1 or v_i=-1, and these vectors get pasted together.
So the vertex (1,-1,1) of the 3-cube would be assigned the vertex

| 1 | 0 | 1 |
|---+---+---|
| 0 | 1 | 0 |

of the hypersimplex.

• When you say y_i + y_{-i} = 1, you also have to specify that this is for all i=1,...,n.

[julian-upc]

• Cubohedral: First, this is an adjective and not a noun; do you want to say "cubohedron"?
• I don't quite understand the definition. Is this for one particular j or for all j? Can you give an example of a cubohedron that is not a cube?
• You say "the edges of C ... identified with the vertices of a subset P". I don't understand what you're trying to do here. Which vertex do you identify an edge of the cube with?

[julian-upc]

Your formula before "Notation" needs some fixing; it should be restriction and not quotient, and y_{-i}, and quantified over all i=1,...,n.

[julian-upc]

Notation: element T of what? What polytope is the height function on?
Definition: hmmmm, we need a better name than "New relation".... how about "cubohedral relation"?