Integral Menger curvature for sets of arbitrary dimension and codimension

We propose a notion of integral Menger curvature for compact, $m$-dimensional sets in $n$-dimensional Euclidean space and prove that finiteness of this quantity implies that the set is $C^{1,\alpha}$ embedded manifold with the H{\"o}lder norm and the size of maps depending only on the curvature. We develop the ideas introduced by Strzelecki and von der Mosel [Adv. Math. 226(2011)] and use a similar strategy to prove our results.