Given a measurable set of positive measure, it is not difficult to show that if and only if is equal to its convex hull minus a set of measure zero. We investigate the stability of this statement: If is small, is close to its convex hull? Our main result is an explicit control, in arbitrary dimension, on the measure of the difference between and its convex hull in terms of .
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Alessio Figalli, David Jerison, Quantitative stability for sumsets in . J. Eur. Math. Soc. 17 (2015), no. 5, pp. 1079–1106DOI 10.4171/JEMS/527