The uniform boundary condition on a normed ch ain complex requires the existence of controlled fillings for all boundari es. The uniform boundary condition naturally comes up in the context of gl ueing results for simplicial volume. Matsumoto and Morita showed that the singular chain complex of spaces with amenable fundamental group satisfies the uniform boundary condition\, using bounded cohomology. We give a dire ct geometric proof of this fact in the aspherical case. This proof admits generalisations to integral foliated simplicial volume\, which provides up per bounds for
