Which kind of integral Menger-type curvature is suitable for a generalization of L\'{e}ger's theorem?

In 1999 J.C. L\'{e}ger proved that a one-dimensional set with finite total Menger curvature is 1-rectifiable. We are trying to generalize this result to higher dimensional sets and curvature energies, where it is not clear how to define those curvature energies. In this talk we give a characterization of integral Menger-type curvatures and a motivation why those may be suitable for this purpose.