\documentclass{article}
\begin{document}
Let $X$ be an $n \times p$ matrix, $Z$ be an $n \times q$ matrix,
$\beta$ be a $p$-vector, $\delta$ be a $q$-vector.  Let $X_{k}$ denote
the $k$th row of $X$ and likewise for $Z_{k}$.

Assuming that $Z$ is not all zeros, is it true that

\begin{equation}
\sum_{i=1}^{n}\sum_{j=1}^{n} |X_{i}\beta + Z_{i}\delta - X_{j}\beta -
Z_{j}\delta| = \sum_{i=1}^{n}\sum_{j=1}^{n} |X_{i}\beta - X_{j}\beta|
\end{equation}
\textbf{if and only if}
\begin{equation}
\delta \equiv 0
\end{equation}
If it is not true in general, under what conditions is it true?  Note
that in a trivial case $|a + b| = |a|$ can be true if $b = 0$ or $b = -2a$.

\end{document}
