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arXiv · 1206.1965

Near equality in the two-dimensional Brunn-Minkowski inequality

Michael Christ
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Abstract

If a pair of subsets of two-dimensional Euclidean space nearly achieves equality in the Brunn-Minkowski inequality, in the sense that the measure of the associated sumset is nearly equal to the lower bound provided by the inequality, then these sets must nearly coincide with a pair of homothetic convex sets. The proof relies on a continuum analogue of a theorem of Freiman which characterizes finite sets of integers whose sumsets are of nearly minimal size. Small corrections and clarifications have been made in this draft.

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Michael Christ. 2012-07-20. Near equality in the two-dimensional Brunn-Minkowski inequality. https://arxiv.org/abs/1206.1965

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