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arXiv · 1412.7586

Quantitative stability for sumsets in $R^n$

Alessio Figalli·David Jerison
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Abstract

Given a measurable set $A\subset \R^n$ of positive measure, it is not difficult to show that $|A+A|=|2A|$ if and only if $A$ is equal to its convex hull minus a set of measure zero. We investigate the stability of this statement: If $(|A+A|-|2A|)/|A|$ is small, is $A$ close to its convex hull? Our main result is an explicit control, in arbitrary dimension, on the measure of the difference between $A$ and its convex hull in terms of $(|A+A|-|2A|)/|A|$.

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Alessio Figalli, David Jerison. 2014-12-24. Quantitative stability for sumsets in $R^n$. https://arxiv.org/abs/1412.7586

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