  The uniform boundary condition in a normed chain complex asks for a uniform
linear bound on fillings of null-homologous cycles. For the $\ell^1$-norm on
the singular chain complex, Matsumoto and Morita established a characterisation
of the uniform boundary condition in terms of bounded cohomology. In
particular, spaces with amenable fundamental group satisfy the uniform boundary
condition in every degree. We will give an alternative proof of statements of
this type, using geometric F{\o}lner arguments on the chain level instead of
passing to the dual cochain complex. These geometric methods have the advantage
that they also lead to integral refinements. In particular, we obtain
applications in the context of integral foliated simplicial volume.
