TY - JOUR

T1 - Inverse additive problems for Minkowski sumsets I

AU - Freiman, G. A.

AU - Grynkiewicz, D.

AU - Serra, O.

AU - Stanchescu, Y. V.

N1 - Funding Information:
D. Grynkiewicz was supported by FWF Grant M1014-N13; O. Serra was supported by the Spanish Research Council MTM2008-06620-C03-01 and the Catalan Research Council 2009SGR01387; The research of Y. Stanchescu was supported by the Open University of Israel’s Research Fund, Grant No. 100937.

PY - 2012/9

Y1 - 2012/9

N2 - We give the structure of discrete two-dimensional finite sets A, B ⊆ ℝ 2 which are extremal for the recently obtained inequality where m and n are the minimum number of parallel lines covering A and B respectively. Via compression techniques, the above bound also holds when m is the maximal number of points of A contained in one of the parallel lines covering A and n is the maximal number of points of B contained in one of the parallel lines covering B. When m, n ≥ 2, we are able to characterize the case of equality in this bound as well. We also give the structure of extremal sets in the plane for the projection version of Bonnesen's sharpening of the Brunn-Minkowski inequality: μ (A + B) ≥ (μ(A)/m + μ(B)/n)(m + n), where m and n are the lengths of the projections of A and B onto a line.

AB - We give the structure of discrete two-dimensional finite sets A, B ⊆ ℝ 2 which are extremal for the recently obtained inequality where m and n are the minimum number of parallel lines covering A and B respectively. Via compression techniques, the above bound also holds when m is the maximal number of points of A contained in one of the parallel lines covering A and n is the maximal number of points of B contained in one of the parallel lines covering B. When m, n ≥ 2, we are able to characterize the case of equality in this bound as well. We also give the structure of extremal sets in the plane for the projection version of Bonnesen's sharpening of the Brunn-Minkowski inequality: μ (A + B) ≥ (μ(A)/m + μ(B)/n)(m + n), where m and n are the lengths of the projections of A and B onto a line.

UR - http://www.scopus.com/inward/record.url?scp=84865744592&partnerID=8YFLogxK

U2 - 10.1007/s13348-012-0060-5

DO - 10.1007/s13348-012-0060-5

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AN - SCOPUS:84865744592

SN - 0010-0757

VL - 63

SP - 261

EP - 286

JO - Collectanea Mathematica

JF - Collectanea Mathematica

IS - 3

ER -