Inverse Additive Problems for Minkowski Sumsets II

The Brunn–Minkowski Theorem asserts that μ _d (A+B)^1/d ≥μ _d (A)^1/d +μ _d (B)^1/d for convex bodies A,B?? ^d , where μ _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 $$mu_d(A+B)geq (M^{1/(d-1)}+N^{1/(d-1)} )^{d-1}biggl(frac{mu_d(A)}{M}+frac {mu_d(B)}{N} biggr),$$ where M=sup?{μ _d?1((x+H)∩A)∣x∈? ^d } and $N=sup{mu_{d-1}((mathbf{y}+H)cap B)mid mathbf{y}in mathbb {R}^{d}}$ . Standard compression arguments show that the above bound also holds when M=μ _d?1(π(A)) and N=μ _d?1(π(B)), where π denotes a projection of ? ^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.