  We give a more elementary proof of a result by Ambrosio, Fusco and Hutchinson
to estimate the Hausdorff dimension of the singular set of minimizers of the
Mumford-Shah energy (see [2, Theorem 5.6]). On the one hand, we follow the
strategy of the above mentioned paper; but on the other hand our analysis
greatly simplifies the argument since it relies on the compactness result
proved by the first two Authors in [4, Theorem 13] for sequences of local
minimizers with vanishing gradient energy, and the regularity theory of minimal
Caccioppoli partitions, rather than on the corresponding results for Almgren's
area minimizing sets.
