I do a lot of my early draft writing in LaTeX, so I place a substantial value in being able to quickly read my own LaTeX code.
As a result, I adopt a style which is half-way between the "LaTeX way" (of using the correct delimiters for math environments and semantic names for macros) and quick-and-dirty macro usage (to save not only on keystrokes made, but on characters to re-read). The following is a typical way that I would render your code-segment.
% In the pre-amble
% ====================
% A semantic command for vectors
\renewcommand\vec[1]{\mathbf{#1}}
% A syntactic command for special symbols which I would
% consistently use for a given purpose.
\newcommand\R{\mathbb{R}}
\newcommand\cS{\mathcal{S}}
% In the document body
% ====================
\subsection{Convex sets}
A set $\cS$ in $\R^n$ is said to be \emph{convex} if
\begin{align}
\vec x_1, \vec x_2, \in \cS
&\implies
\lambda \vec x_1 + (1-\lambda) \vec x_2 \in \cS
&
\text{for all $0 < \lambda < 1$.}
\end{align}
I don't bother adding newlines in the middle of sentences, except for things such as footnotes and ends of lines (which is useful to do if you use a content revision system to track changes). But I add them liberally in my displayed math to better reveal the structure of what I'm writing, especially in tabulated environments such as align
. You could also just use an equation
environment instead, in which case I would write:
\begin{equation}
\vec x_1, \vec x_2, \in \cS
\implies
\lambda \vec x_1 + (1-\lambda) \vec x_2 \in \cS
\quad
\text{for all $0 < \lambda < 1$.}
\end{equation}
Some further remarks on my approach to defining the macros:
If you only have a few vectors in your text, you can make your mathematics somewhat more terse by defining
\newcommand\vx{\vec x}
to speed up your (or your collaborators') personal lexing process; how this affects one's ability to parse the mathematics is another matter. As with \cS
, this sort of naming convention depends on the reader (most probably just you, but also any collaborators) becoming accustomed to parsing the name of the macro as an adjective-noun couplet: a "calligraphic S", a "vector x", and so on.
As for \R
, this sort of naming convention ought only to be used for a single style thoughout all of your documents for symbols representing very important objects (such as blackboard-bold symbols denoting number-sets which you are likely to refer to in any given paper).
Use such shortcuts with discretion, both for your own sanity and those of your colleagues who might also work on the document with you.
\quad\text{for all}\quad
instead of\text{~~~for all~~~}
and if you use it a lot, define\newcommand\qtext[1]{\quad\text{#1}\quad}
and possibly\newcommand\qqtext[1]{\qquad\text{#1}\qquad}
if you use larger space somewhere as well.