  We show that the theory of varifolds can be suitably enriched to open the way
to applications in the field of discrete and computational geometry. Using
appropriate regularizations of the mass and of the first variation of a
varifold we introduce the notion of approximate mean curvature and show various
convergence results that hold in particular for sequences of discrete varifolds
associated with point clouds or pixel/voxel-type discretizations of
$d$-surfaces in the Euclidean $n$-space, without restrictions on dimension and
codimension. The variational nature of the approach also allows to consider
surfaces with singularities, and in that case the approximate mean curvature is
consistent with the generalized mean curvature of the limit surface. A series
of numerical tests are provided in order to illustrate the effectiveness and
generality of the method.
