Quantitative stability for sumsets in $R^n$

By Alessio Figalli, David Jerison. 2014

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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 $(|A+A|-|2A|)/|A|$.

Subfields: Applied Mathematics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/b2d894b23b756caa80c2f88fbc836ce21b01015c