  We consider the thresholding scheme and explore its connection to De Giorgi's
ideas on gradient flows in metric spaces; here applied to mean curvature flow
as the steepest descent of the interfacial area.
  The basis of our analysis is the observation by Esedoglu and the second
author that thresholding can be interpreted as a minimizing movements scheme
for an energy that approximates the interfacial area.
  De Giorgi's framework provides an optimal energy dissipation relation for the
scheme in which we pass to the limit to derive a dissipation-based weak
formulation of mean curvature flow.
  Although applicable in the general setting of arbitrary networks, here we
restrict ourselves to the case of a single interface, which allows for a
compact, self-contained presentation.
