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 Bourgains 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.
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