  We introduce and study finite $d$-volumes - the high dimensional
generalization of finite metric spaces. Having developed a suitable
combinatorial machinery, we define $\ell_1$-volumes and show that they contain
Euclidean volumes and hypertree volumes. We show that they can approximate any
$d$-volume with $O(n^d)$ multiplicative distortion. On the other hand, contrary
to Bourgain's theorem for $d=1$, there exists a $2$-volume that on $n$ vertices
that cannot be approximated by any $\ell_1$-volume with distortion smaller than
$\tilde{\Omega}(n^{1/5})$.
  We further address the problem of $\ell_1$-dimension reduction in the context
of $\ell_1$ volumes, and show that this phenomenon does occur, although not to
the same striking degree as it does for Euclidean metrics and volumes. In
particular, we show that any $\ell_1$ metric on $n$ points can be $(1+
\epsilon)$-approximated by a sum of $O(n/\epsilon^2)$ cut metrics, improving
over the best previously known bound of $O(n \log n)$ due to Schechtman.
  In order to deal with dimension reduction, we extend the techniques and ideas
introduced by Karger and Bencz{\'u}r, and Spielman et al.~in the context of
graph Sparsification, and develop general methods with a wide range of
applications.
