  Good approximations have been attained for the sparsest cut problem by
rounding solutions to convex relaxations via low-distortion metric embeddings.
Recently, Bryant and Tupper showed that this approach extends to the hypergraph
setting by formulating a linear program whose solutions are so-called
diversities which are rounded via diversity embeddings into $\ell_1$.
Diversities are a generalization of metric spaces in which the nonnegative
function is defined on all subsets as opposed to only on pairs of elements.
  We show that this approach yields a polytime $O(\log{n})$-approximation when
either the supply or demands are given by a graph. This result improves upon
Plotkin et al.'s $O(\log{(kn)}\log{n})$-approximation, where $k$ is the number
of demands, for the setting where the supply is given by a graph and the
demands are given by a hypergraph. Additionally, we provide a polytime
$O(\min{\{r_G,r_H\}}\log{r_H}\log{n})$-approximation for when the supply and
demands are given by hypergraphs whose hyperedges are bounded in cardinality by
$r_G$ and $r_H$ respectively.
  To establish these results we provide an $O(\log{n})$-distortion $\ell_1$
embedding for the class of diversities known as diameter diversities. This
improves upon Bryant and Tupper's $O(\log\^2{n})$-distortion embedding. The
smallest known distortion with which an arbitrary diversity can be embedded
into $\ell_1$ is $O(n)$. We show that for any $\epsilon > 0$ and any $p>0$,
there is a family of diversities which cannot be embedded into $\ell_1$ in
polynomial time with distortion smaller than $O(n^{1-\epsilon})$ based on
querying the diversities on sets of cardinality at most $O(\log^p{n})$, unless
$P=NP$. This disproves (an algorithmic refinement of) Bryant and Tupper's
conjecture that there exists an $O(\sqrt{n})$-distortion $\ell_1$ embedding
based off a diversity's induced metric.
