  Let h : R $\rightarrow$ R+ be a lower semi-continuous subbadditive and even
function such that h(0) = 0 and h($\theta$) $\ge$ $\alpha$|$\theta$| for some
$\alpha$ > 0. The h-mass of a k-polyhedral chain P =$\sum$j $\theta$j$\sigma$j
in R n (0 $\le$ k $\le$ n) is defined as M h (P) := j h($\theta$j) H k
($\sigma$j). If T = $\tau$ (M, $\theta$, $\xi$) is a k-rectifiable chain, the
definition extends to M h (T) := M h($\theta$) dH k. Given such a rectifiable
flat chain T with M h (T) < $\infty$ and $\partial$T polyhedral, we prove that
for every $\eta$ > 0, it decomposes as T = P + $\partial$V with P polyhedral, V
rectifiable, M h (V) < $\eta$ and M h (P) < M h (T) + $\eta$. In short, we have
a polyhedral chain P which strongly approximates T in h-mass and preserves the
homological constraint $\partial$P = $\partial$T. These results are motivated
by the study of approximations of M h by smoother functionals but they also
provide explicit formulas for the lower semicontinuous envelope of T
$\rightarrow$ M h (T) + I $\partial$S ($\partial$T) with respect to the
topology of the flat norm.
