On the lower semicontinuous envelope of functionals defined on polyhedral chains

In this note we prove an explicit formula for the lower semicontinuous envelope of some functionals defined on real polyhedral chains. More precisely, denoting by H : R -> [0,infinity) an even, subadditive, and lower semicontinuous function with H(0) = 0, and by Phi(H) the functional induced by H on polyhedral m-chains, namely Phi(H) (P) := Sigma(N)(i=1) H(theta(i))H-m(sigma(i)), for every P = Sigma(N)(i=1)theta(i) [[sigma(i)]] is an element of P-m(R-n), we prove that the lower semicontinuous envelope of Phi(H) coincides on rectifiable m-currents with the H-mass M-H (R) := integral(E) H(theta(x)) dH(m)(x) for every R = [[E, tau, theta]] is an element of R-m(R-n).