Separate long math text under sum symbol into different lines?

You can use the \substack command from the amsmath package, like this:

\begin{equation}
  d(\vec{x},\vec{y}) =
  \sum_{\substack{Z_{xy}\in\vec{Z}_{xy}\\
                  \forall x\in\vec{x}\\
                  \forall y\in\vec{y}}}
        f(Z_{xy}) 
\end{equation}

However, the result still doesn’t look good, because of the extra spacing around the sum symbol:

To fix this, you can use the \mathclap command from the mathtools package, like this:

\begin{equation}
 d(\vec{x},\vec{y}) =
    \sum_{\mathclap{\substack{Z_{xy}\in\vec{Z}_{xy}\\
                              \forall x\in\vec{x}\\
                               \forall y\in\vec{y}}}}
          f(Z_{xy}) 
\end{equation}

But perhaps you might be happy using only \mathclap, and not \substack. The result looks good as long as the subscript is not too wide.

\begin{equation}
  d(\vec{x},\vec{y}) =
    \sum_{\mathclap{{Z_{xy}\in\vec{Z}_{xy},
          \forall x\in\vec{x},
          \forall y\in\vec{y}}}} f(Z_{xy})
\end{equation}

The mathtools package also have several other useful commands for typesetting mathematics, including more commands for improving the display of subscripts and superscripts. I very much recommend taking a look at its documentation.

Try the \substack command from the amsmath package, details of which are found here

I think it can be done without \substack as

\sum_{a=b \\ b=c \\ c=a}

wich results in