Inverse Additive Problems for Minkowski Sumsets II

The Brunn-Minkowski Theorem asserts that mu(d)(A+B)(1/d) >=mu(d)(A)(1/d) + mu(d)(B)(1/d) for convex bodies A, B subset of R-d , where mu(d) denotes the d-dimensional Lebesgue measure. It is well known that equality holds if and only if A and B are homothetic, but few characterizations of equality in other related bounds are known. Let H be a hyperplane. Bonnesen later strengthened this bound by showing mu(d)(A+B)(1/d) >=(M1/(d-1)+N1/(d-1))(d-1) (mu(d)(A)/M + mu(d)(B)/N), where M = sup{mu(d-1)((x + H)boolean AND A) vertical bar x is an element of R-d} and N = sup{mu(d-1)((y + H)boolean AND B) vertical bar y is an element of R-d}. Standard compression arguments show that the above bound also holds when M = mu(d-1)(pi(A)) and N = mu (d-1)(pi(B)), where pi denotes a projection of R-d onto H, which gives an alternative generalization of the Brunn-Minkowski bound. In this paper, we characterize the cases of equality in this latter bound, showing that equality holds if and only if A and B are obtained from a pair of homothetic convex bodies by 'stretching' along the direction of the projection, which is made formal in the paper. When d = 2, we characterize the case of equality in the former bound as well.