A Convex Decomposition Formula for the Mumford-Shah Functional in Dimension One

We study the convex lift of Mumford-Shah type functionals in the space of rectifiable currents and we prove a convex decomposition formula in dimension one, for finite linear combinations of SBV graphs. We use this result to prove the equivalence between the minimum problems for the Mumford-Shah functional and the lifted one and, as a consequence, we obtain a weak existence result for calibrations in one dimension.