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 $H \colon \mathbb{ R} \to \left[ 0,\infty \right)$ 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 \[ \Phi_{ H}(P) := \sum_{ i=1}^{ N} H(\theta_{ i}) \mathcal{ H}^{ m}(\sigma_{ i}), \quad\mbox{ for every }P=\sum_{ i=1}^{ N} \theta_{ i} [[ \sigma_{ i} ]] \in\mathbf{ P}_m(\mathbb{ R}^n), \] we prove that the lower semicontinuous envelope of $\Phi_H$ coincides on rectifiable $m$-currents with the $H$-mass \[ \mathbb{ M}_{ H}(R) := \int_E H(\theta(x)) \, d\mathcal{ H}^m(x) \quad \mbox{ for every } R= [[ E,\tau,\theta ]] \in \mathbf{ R}_{ m}(\mathbb{ R}^{ n}). \]
Categories: math.AP