  For graphs there exists a strong connection between spectral and
combinatorial expansion properties. This is expressed, e.g., by the discrete
Cheeger inequality, the lower bound of which states that $\lambda(G) \leq
h(G)$, where $\lambda(G)$ is the second smallest eigenvalue of the Laplacian of
a graph $G$ and $h(G)$ is the Cheeger constant measuring the edge expansion of
$G$. We are interested in generalizations of expansion properties to finite
simplicial complexes of higher dimension (or uniform hypergraphs).
  Whereas higher dimensional Laplacians were introduced already in 1945 by
Eckmann, the generalization of edge expansion to simplicial complexes is not
straightforward. Recently, a topologically motivated notion analogous to edge
expansion that is based on $\mathbb{Z}_2$-cohomology was introduced by Gromov
and independently by Linial, Meshulam and Wallach. It is known that for this
generalization there is no higher dimensional analogue of the lower bound of
the Cheeger inequality. A different, combinatorially motivated generalization
of the Cheeger constant, denoted by $h(X)$, was studied by Parzanchevski,
Rosenthal and Tessler. They showed that indeed $\lambda(X) \leq h(X)$, where
$\lambda(X)$ is the smallest non-trivial eigenvalue of the ($(k-1)$-dimensional
upper) Laplacian, for the case of $k$-dimensional simplicial complexes $X$ with
complete $(k-1)$-skeleton.
  Whether this inequality also holds for $k$-dimensional complexes with
non-complete $(k-1)$-skeleton has been an open question. We give two proofs of
the inequality for arbitrary complexes. The proofs differ strongly in the
methods and structures employed, and each allows for a different kind of
additional strengthening of the original result.
