Inverse Additive Problems for Minkowski Sumsets II

The Brunn-Minkowski Theorem asserts that \(\mu_d(A+B)^{1/d}\geq \mu_d(A)^{1/d}+\mu_d(B)^{1/d}\) for convex bodies \(A,\,B\subseteq \R^d\), 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 $$\mu_d(A+B)\geq (M^{1/(d-1)}+N^{1/(d-1)})^{d-1}(\frac{\mu_d(A)}{M}+\frac{\mu_d(B)}{N}),$$ where \(M=\sup\{\mu_{d-1}((\mathbf x+H)\cap A)\mid \mathbf x\in \R^d\}\) and \(N=\sup\{\mu_{d-1}((\mathbf y+H)\cap B)\mid \mathbf y\in \R^d\}\). 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 \(\mathbb R^d\) onto \(H\), which gives an alternative generalization of the Brunn-Minkowski bound. In this paper, we characterize the cases of equality in this later 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.