Inverse Additive Problems for Minkowski Sumsets II

The Brunn–Minkowski Theorem asserts that μd(A+B)1/d ≥ μd(A) + μd(B) 1/d for convex bodies A,B ⊆ R , where μd 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 ) , Communicated by Marco Abate. D.G. was supported by the FWF Austrian Science Fund Project P21576-N18. O.S. was supported by the Catalan Research Council under project 2009 SGR: 1387 and by the Spanish Research Council under project MTM2008-06620-C03-01. G.A. Freiman The Raymond and Beverly Sackler Faculty of Exact Sciences School of Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel e-mail: [email protected] D. Grynkiewicz ( ) Institut für Mathematik und Wissenschaftliches Rechnen, Karl-Franzens-Universität, Graz, Austria e-mail: [email protected] O. Serra Departament de Matemàtica Aplicada IV, Universitat Politècnica de Catalunya, Barcelona, Spain e-mail: [email protected] Y.V. Stanchescu The Open University of Israel, Raanana 43107, Israel e-mail: [email protected] Y.V. Stanchescu Afeka Academic College, Tel Aviv 69107, Israel e-mail: [email protected]