  We prove a new quantitative version of the Alexandrov theorem which states
that if the mean curvature of a regular set in R^{n+1} is close to a constant
in L^{n}-sense, then the set is close to a union of disjoint balls with respect
to the Hausdorff distance. This result is more general than the previous
quantifications of the Alexandrov theorem and using it we are able to show that
in R^2 and R^3 a weak solution of the volume preserving mean curvature flow
starting from a set of finite perimeter asymptotically convergences to a
disjoint union of equisize balls, up to possible translations. Here by weak
solution we mean a flat flow, obtained via the minimizing movements scheme.
