# Quantitative stability for sumsets in $\mathbb R^n$

### Alessio Figalli

ETH Zürich, Switzerland### David Jerison

Massachusetts Institute of Technology, Cambridge, USA

## Abstract

Given a measurable set $A\subset \mathbb R^n$ 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 $(|A+A|-|2A|)/|A|$.

## Cite this article

Alessio Figalli, David Jerison, Quantitative stability for sumsets in $\mathbb R^n$. J. Eur. Math. Soc. 17 (2015), no. 5, pp. 1079–1106

DOI 10.4171/JEMS/527