Inverse additive problems for Minkowski sumsets I

G. A. Freiman, D. Grynkiewicz, O. Serra*, Y. V. Stanchescu

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

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.

Original languageEnglish
Pages (from-to)261-286
Number of pages26
JournalCollectanea Mathematica
Volume63
Issue number3
DOIs
StatePublished - Sep 2012

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