In this paper, we prove short time existence and uniqueness of smooth evolution by mean curvature in Rn+1 starting from any n-dimensional."; (epsilon, R)-Reifenberg flat set with epsilon sufficiently small. More precisely, we show that the level set flow in such a situation is non-fattening and smooth. These sets have a weak metric notion of tangent planes at every small scale, but the tangents are allowed to tilt as the scales vary. As for every n this class is wide enough to include some fractal sets, we obtain unique smoothing by mean curvature flow of sets with Hausdorff dimension larger than n, which are additionally not graphical at any scale. Except in dimension one, no such examples were previously known.
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