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Quantitative stability for sumsets in \(\mathbb R^n\). (English) Zbl 1325.49052

Summary: 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 \((|A+A|-|2A|)/|A|\).

MSC:

49Q20 Variational problems in a geometric measure-theoretic setting
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References:

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