  A pair of subsets of Euclidean space which nearly achieves equality in the
Brunn-Minkowski inequality must nearly coincide with a pair of homothetic
convex sets. The two-dimensional case was treated in a previous paper in this
series by an argument which does not seem to generalize to higher dimensions.
Here the result is extended to arbitrary dimensions. An induction on the
dimension, a symmetrization argument, and a description of near solutions of an
additive functional equation are used to establish sufficient regularity to set
up a compactness argument.
