  If a pair of subsets of two-dimensional Euclidean space nearly achieves
equality in the Brunn-Minkowski inequality, in the sense that the measure of
the associated sumset is nearly equal to the lower bound provided by the
inequality, then these sets must nearly coincide with a pair of homothetic
convex sets. The proof relies on a continuum analogue of a theorem of Freiman
which characterizes finite sets of integers whose sumsets are of nearly minimal
size.
  Small corrections and clarifications have been made in this draft.
