Geodesic
Quantitative stability for sumsets in $\mathbb R^n$
Journal of the European Mathematical Society, Tome 17 (2015) no. 5, pp. 1079-1106 Cet article a éte moissonné depuis la source EMS Press

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Given a measurable set A⊂Rn 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∣.
DOI : 10.4171/jems/527
Classification : 49-XX, 34-XX
Keywords: Quantitative stability, sumsets, Freiman’s theorem 
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     author = {Alessio Figalli and David Jerison},
     title = {Quantitative stability for sumsets in $\mathbb R^n$},
     journal = {Journal of the European Mathematical Society},
     pages = {1079--1106},
     year = {2015},
     volume = {17},
     number = {5},
     doi = {10.4171/jems/527},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/jems/527/}
}
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Alessio Figalli; David Jerison. Quantitative stability for sumsets in $\mathbb R^n$. Journal of the European Mathematical Society, Tome 17 (2015) no. 5, pp. 1079-1106. doi: 10.4171/jems/527

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