  Cheeger's fundamental inequality states that any edge-weighted graph has a
vertex subset $S$ such that its expansion (a.k.a. conductance) is bounded as
follows: \[ \phi(S) \defeq \frac{w(S,\bar{S})}{\min \set{w(S), w(\bar{S})}}
\leq 2\sqrt{\lambda_2} \] where $w$ is the total edge weight of a subset or a
cut and $\lambda_2$ is the second smallest eigenvalue of the normalized
Laplacian of the graph. Here we prove the following natural generalization: for
any integer $k \in [n]$, there exist $ck$ disjoint subsets $S_1, ..., S_{ck}$,
such that \[ \max_i \phi(S_i) \leq C \sqrt{\lambda_{k} \log k} \] where
$\lambda_i$ is the $i^{th}$ smallest eigenvalue of the normalized Laplacian and
$c<1,C>0$ are suitable absolute constants. Our proof is via a polynomial-time
algorithm to find such subsets, consisting of a spectral projection and a
randomized rounding. As a consequence, we get the same upper bound for the
small set expansion problem, namely for any $k$, there is a subset $S$ whose
weight is at most a $\bigO(1/k)$ fraction of the total weight and $\phi(S) \le
C \sqrt{\lambda_k \log k}$. Both results are the best possible up to constant
factors.
  The underlying algorithmic problem, namely finding $k$ subsets such that the
maximum expansion is minimized, besides extending sparse cuts to more than one
subset, appears to be a natural clustering problem in its own right.
