naa a22 4500 51081039X CHVBK 20180411083439.0 cr unu---uuuuu 180411e20130101xx s 000 0 eng 10.1007/s12220-011-9251-7 doi (NATIONALLICENCE)springer-10.1007/s12220-011-9251-7 Inverse Additive Problems for Minkowski Sumsets II [Elektronische Daten] [G. Freiman, D. Grynkiewicz, O. Serra, Y. Stanchescu] 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 $$\mu_d(A+B)\geq (M^{1/(d-1)}+N^{1/(d-1)} )^{d-1}\biggl(\frac{\mu_d(A)}{M}+\frac {\mu_d(B)}{N} \biggr),$$ where M=sup {μ d−1((x+H)∩A)∣x∈ℝ 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 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. Mathematica Josephina, Inc., 2011 Brunn-Minkowski nationallicence Convex bodies nationallicence Sumset nationallicence Convex functions nationallicence Freiman G. The Raymond and Beverly Sackler Faculty of Exact Sciences School of Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel aut Grynkiewicz D. Institut für Mathematik und Wissenschaftliches Rechnen, Karl-Franzens-Universität, Graz, Austria aut Serra O. Departament de Matemàtica Aplicada IV, Universitat Politècnica de Catalunya, Barcelona, Spain aut Stanchescu Y. The Open University of Israel, 43107, Raanana, Israel aut Journal of Geometric Analysis Springer-Verlag 23/1(2013-01-01), 395-414 1050-6926 23:1<395 2013 23 12220 https://doi.org/10.1007/s12220-011-9251-7 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s12220-011-9251-7 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Freiman G. The Raymond and Beverly Sackler Faculty of Exact Sciences School of Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel aut NATIONALLICENCE 700 1- Grynkiewicz D. Institut für Mathematik und Wissenschaftliches Rechnen, Karl-Franzens-Universität, Graz, Austria aut NATIONALLICENCE 700 1- Serra O. Departament de Matemàtica Aplicada IV, Universitat Politècnica de Catalunya, Barcelona, Spain aut NATIONALLICENCE 700 1- Stanchescu Y. The Open University of Israel, 43107, Raanana, Israel aut NATIONALLICENCE 773 0- Journal of Geometric Analysis Springer-Verlag 23/1(2013-01-01), 395-414 1050-6926 23:1<395 2013 23 12220 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer