Wider space between minus sign and number when surrounded by verts

Question

When I write the following:

\documentclass[12pt,a4paper]{article} 
\begin{document}  

$$
-1, -2, -3, \dots \| -1 \|, \| -2 \|, \| -3 \|
$$

\end{document}  

It produces the following:

As you can see, the negative signs are more closely coupled with the constants in the first three numbers, but this fails to be the case when surrounded by norm signs, which I find aesthetically displeasing. How can I fix this?

Answer 1

As you've discovered, the symbols generated by \vert (equivalently: |) and \Vert (equivalently: \|) have status "math-ordinary". Hence, TeX interprets the - symbol as a binary operator, since the - symbol is sandwiched between two symbols (\Vert and a numeral) with status "math-ordinary". (This is, of course, the correct default behavior for expressions such as $a-b$.) To get TeX to treat the - symbol as a unary operator, it's advisable to use \lVert and \rVert, which have status "math-open" and "math-close", respectively, instead of just \Vert.

Better still, define a LaTeX macro called, say, \norm, which automatically uses the correct math status values for the opening and closing fences. In the code below, the macro \norm is set up in such a way that \norm* is defined as well; the latter lets the size of the "fences" grow automatically, as needed.

\documentclass[12pt,a4paper]{article} 
\usepackage{mathtools} % for '\DeclarePairedDelimiter' macro
\DeclarePairedDelimiter{\abs}{\lvert}{\rvert}
\DeclarePairedDelimiter{\norm}{\lVert}{\rVert}

\begin{document}  

Original form:

$
-1, -2, -3, \dots, \| -1 \|, \| -2 \|, \| -3 \|
$

\medskip
Better:

$
-1, -2, -3, \dots, \lVert -1 \rVert, \lVert -2 \rVert, \lVert -3 \rVert
$

\medskip
Best:

$
-1, -2, -3, \dots, \norm{-1}, \norm{-2}, \norm{-3}
$

\end{document}  

Answer 2

The minus sign sees the left delimiter as the first argument of a subtraction operation and sets the space as such. Enclosing the - in braces forces it to treat the minus as a unary operator (a negation on the following number). (Likewise, I could enclose the whole number in braces as \| {-1} \|)

\documentclass{article} 
\begin{document}  

\[
  -1, -2, -3, \dots \| {-}1 \|, \| {-}2 \|, \| {-}3 \|
\]

\end{document}