Near equality in the two-dimensional Brunn-Minkowski inequality

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.

References in corpus (4)

• Near Equality in the Brunn-Minkowski Inequality
• An Approximate Inverse Riesz-Sobolev Inequality
• Near-extremizers of Young's Inequality for R^d
• Near-extremizers of Young's inequality for discrete groups

Cited by in corpus (9)

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• Near equality in the Riesz-Sobolev inequality in higher dimensions
• On an extremization problem concerning Fourier coefficients
• Quantitative Borell-Brascamp-Lieb inequalities for compactly supported power concave functions (and some applications)
• Quantitative stability for sumsets in $R^n$
• A constrained optimization problem for the Fourier transform: Existence