Inverse additive problems for Minkowski Sumsets II

Abstract

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⊂R 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 μd (A + B)≥ (M1/(d-1) + N1/(d-1))d-1 (μd (A)/M+μd(B)/N), where M=sup {μ d-1((x+H)⊂A) x ∈ R d} and N=sup{μ d-1 (y+H)∩B)∥ y∈Rd. 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 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. © 2011 Mathematica Josephina, Inc.

Publication Title

Journal of Geometric Analysis

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